Research ArticleA Generalized National Planning Approach forAdmission Capacity in Higher Education: A NonlinearInteger Goal Programming Model with a Novel DifferentialEvolution Algorithm
Said Ali El-Qulity1 and Ali Wagdy Mohamed2
1Department of Industrial Engineering, Faculty of Engineering, KingAbdulaziz University, P.O. Box 80200, Jeddah 21589, Saudi Arabia2Department of Operations Research, Institute of Statistical Studies and Research, Cairo University, Giza 12613, Egypt
Correspondence should be addressed to Ali Wagdy Mohamed; [email protected]
Received 23 May 2015; Revised 27 August 2015; Accepted 3 September 2015
This paper proposes a nonlinear integer goal programming model (NIGPM) for solving the general problem of admission capacityplanning in a country as a whole.The work aims to satisfy most of the required key objectives of a country related to the enrollmentproblem for higher education. The system general outlines are developed along with the solution methodology for application tothe time horizon in a given plan. The up-to-date data for Saudi Arabia is used as a case study and a novel evolutionary algorithmbased on modified differential evolution (DE) algorithm is used to solve the complexity of the NIGPM generated for different goalpriorities. The experimental results presented in this paper show their effectiveness in solving the admission capacity for highereducation in terms of final solution quality and robustness.
1. Introduction
One of the key transformations in global higher education(HE) is the rapid growth of the sector. Growth started in thelast four or five decades of the 20th century and continuesafter the turn of the century. Worldwide, the number ofstudents in higher education has increased from 98 millionin 2000 to over 150 million in 2007, implying a growth ofover 50% in less than ten years. Worldwide gross enrolmentratios (defined as the total enrolment in HE, regardless ofage, expressed as a percentage of the eligible official school-age population corresponding to the same level of educationin a given school year) in the same period show an increasefrom 19% to 26% [1]. According toTrow [2],HE systemsmovefrom elite through mass to universal HE. Elite systems arecharacterized by low enrolment rates (between 0 and 15%).In systems ofmassHE, themain function of higher educationis transmission of skills and preparation for a broader range
of technical and economic elite roles. Access is a right forthose with certain qualifications and enrolment rates varybetween 16 and 50%. Finally, universal HE is characterizedby enrolment rates larger than 50%. In these systems, accessto HE is perceived as an obligation for the middle and upperclasses and the function of HE is related to adaptation of“whole population” to rapid social and technological change.
A growing demand for higher education requires on thesupply side balanced growth in staff, both academic andadministrative, and in facilities and infrastructure. However,growth in the supply of HE often is hampered by competitionon the labor market for qualified personnel. Ashcroft andRayner [3] indicate that, particularly, graduates with higherdegrees are also in demand by the private and governmentsector.
The situation is aggravated when, as often is the case,the income is not keeping pace with the growth in stu-dent numbers. Without sufficient investments in facilities
Hindawi Publishing CorporationComputational Intelligence and NeuroscienceVolume 2016, Article ID 5207362, 14 pageshttp://dx.doi.org/10.1155/2016/5207362
2 Computational Intelligence and Neuroscience
and infrastructure, institutions are left with “inadequateresources for books and journals, equipment, computers, andtelecommunications” [4]. Furthermore, lack of funding leadsto an increase in student staff ratios creating situations inwhich “students literally are unable to find room in classes”[5]. Rising social aspirations and growing socioeconomicrelevance lead also to demands for increased performanceof High Education in interrelated areas for increase in thelabor market relevance of education and the supply of a morediversified student specialty suitable for the market needs.However, numerous reports indicate mismatches betweensupply and demand of graduates [6]. Lack of responsivenessof education systems to new labor market demands willhinder the development since research, for example, indicatesthat, in countries with more engineering students, the econ-omy grows faster than in countries with more lawyers [7].
The Ministry of Higher Education in the Kingdom ofSaudi Arabia is keen on working out its strategic plans andensuring their compatibility with the government’s develop-ment plan. To this effect, the ministry has put in perspectivea number of vital objectives in its ninth five-year plan andthe horizon plan for higher education (AAFAQ, 2014) [8]while attempting to benefit from the international trendsby attracting international experts in the field of highereducation strategic planning (Ministry of Higher Education,2010) [9]. The ministry launched an initiative to preparea modern and a long-term plan for university educationto meet challenges of high population growth rate, ever-increasing funding demands, labor market needs for highlyqualified graduates and student-to-faculty ratio, and so forth[10].
The higher education authority in any country alwaysraises the question of admission capacity for the highereducation institutions and how it is able to respond to thevarious challenges it is facing. The tradition way for tacklingsuch a problem is to design separate plans that align with thespecific needs and behaviors of each individual university tocope with the available resources and capacity.
The current paper presents amore generalizedmathemat-ical model for higher education sector that can successfullymeet the national social, economic, and cultural challengesthat face higher education admission capacity problems overthe coming years. The model is general that it can be appliedfor different countries and/or universities.
It is known thatGP is an extension of linear programminginvolving an objective function with multiple objectives [11].The traditional GP model can be easily solved by simplexmethod or by using many computerized software programssuch as Microsoft Excel Solver add-on and the LINGOpackage [12]. However, it should be noted that there aremany other types of GP models that may include large-scaleand nonlinear relations; such models with large number ofinteger variables add a computational challenge and extralevel of difficulty for solving using classical programmingtechniques. Consequently, using metaheuristic techniques asa substitute for traditional programming methods in orderto solve hard GP problems is an open research area [13,14]. Thus, due to the complexity of the proposed model, anovel evolutionary algorithm based on modified constrained
differential evolution algorithm is developed to solve theproposed nonlinear integer GP model.
The paper is organized as follows. Section 2 handledthe literature review for the problem under study. Section 3explained the general outlines of the system including all theinputs together with the components of the mathematicalmodel and its objectives. Section 4 explains the solutionmethodology over the planning horizon. Section 5 showedhow the proposed NIGPM is developed to adapt generalproposed goals for a country. Section 6 utilized the proposedNIGPM for Saudi Arabia as a case. The solution of the modeland discussions are given as well. The proposed differentialevolution approach and problem solutions were explained inSection 7. The conclusions and points for future researchesare summarized in Section 8.
2. Literature Review
Over the last four decades, a variety of optimization methodshave been developed to solve university admissions planningsystem. Virtually, one of the most effective and powerfulmathematical programming techniques to formulate andsolve optimization problems, proposed in the literature, isgoal programming (GP) technique. A considerable numberof research studies have been proposed to optimize universityadmission planning problems by using goal programmingtechnique alone or combined with other classical mathe-matical methods or intelligent optimization algorithms. Infact, many academicians and researchers have embraced thistechnique as an appropriate technique in such optimizationproblem. The main reason of using GP is its capabilityof simultaneously satisfying several conflicting goals withvarying priorities relevant to the decision-making situation.GP approach was firstly proposed by Lee and Clayton [15],for an optimum allocation of resources in institutions ofhigher learning. The scope of this study was limited to theplanning of one college within the university. Additionally,the planning horizon was also limited to one year. Thismodel was based on actual operational data at the College ofBusiness, Virginia Polytechnic Institute and State University,United States of America. Many types of constraints andvariables regarding the academic staff only were taken intoconsideration in this study such as total number of academicstaff, distribution of academic staff, and number of gradu-ate research assistants. However, all other remaining issuesregarding academic process at universities were excluded.Likewise, Schroeder [16] introduced a new approach forrecourse planning in the universities based on GP. Data weregathered at the University of Minnesota, Minnesota State,United States of America, representing three large academicdepartments over a three-year period. The goals were facultyinstruction loads, staff-to-faculty ratios, faculty distributionby rank, and teaching-assistant-to-faculty ratios. These spec-ified goals are achieved as closely as possible, subject toconstraints on the projected budget available in each yearof the planning horizon and to faculty-flow constraints. Thedecision variables are the faculty, staff, and teaching-assistantlevels in each of the several academic units over the planning
Computational Intelligence and Neuroscience 3
horizon.Themodel was used for long-range budget planningand resource allocation. On the other hand, Lee and Moore[17] developed a goal programming model to determine thebasic composition of the total group of new students tobe admitted into an educational institution. The data wereobtained from a land-grant university; it is located in asoutheastern state, United States of America. It determinesthe number of students that should be admitted in each ofthe various categories but does not specify which individualstudents should be offered admission. They proposed a verysimple admissions planningmodel.Thedecision variables aredefined as the number of admitted students in each categoryof in-state or out-of-state students; freshman, transfer, orreadmitted students; and men or women students and alltheir possible combinations. Thus, they focused on formu-lating an admission policy for the newly entering studentsonly. In 1981, Kendall and Luebbe [18] developed a GPmodel to manage recruitment activities in the small four-year Concordia College in Nebraska. Their model identifiesthe type and number of activities that must be completedeach quarter in order to reach an enrollment goal for a givenyear. These activities included budget, time, manpower, andmarketing strategies. The goal was to enable recruiters tomeet enrollments while managing recruiting resources andactivities in order to remain within the recruiting budget.They concentrated on university financial related problemsfor private colleges. Soyibo and Lee [19] developed a large-scale GP model for efficient resource allocation for IbadanUniversity, Nigeria, which includes eight faculties and acollege of medicine over a five-year planning horizon. Thismodel defines student enrollment and academic staff levelgoals. Linear programming techniques have been also usedin higher education planning. In 2009, Khan [20] used aproduct mix model of linear programming for university’soptimal enrollment management. The aim is to have the besttuition contribution to the campus using the best studentmix and optimal use of those constraints that impact studentenrollment every semester. In 2013, Kassa [21] used a linearprogramming approach for placement of applicants to studyprograms developed and implemented at the College ofBusiness & Economics, Bahir DarUniversity, in Ethiopia.Theapproach is estimated to significantly streamline the place-ment decision process at the college by reducing requiredman-hour as well as the time it takes to announce placementdecisions. Decision support system has been proposed asa new trend of research in higher education in univer-sity planning system as the academic financial planning,admission policy, resource allocation, recruiting, managinguniversity fund, budgeting, and classroom scheduling havebecome a highly complex system with huge data bases.Therefore, several attempts in developing DSS to deal withone or more of these subsystems have been done. Resourceallocation in a university received considerable attentionfrom several authors. A goal programming-based DSS hasbeen presented by Franz et al. [22]. In this approach,they attempted to adopt a variety of academic decision-makers, with differing planning views in an environmentof multiple conflicting objectives. They report that testingof their DSS on four academic decision-makers in large
US Midwestern University shows considerable promise forsupporting decision-makers with varied problem-solvingstyles. Similarly, to find the optimal admission policy, aDSS for student admission policy to Kuwait University,Kuwait, has been developed by Eliman [23]. The DSS iscomposed of three modeling components: first, the academicperformance analysis model which consists of two parts (amulticlassification analysis (MCA) and cohort analysis); sec-ond, models to estimate secondary school graduates supplyand demand for university graduates using demographicgrowth and regression analysis; third, student allocationmodels that use a linear programming formulation. Theoverall reaction of the decision-makers in Kuwait Universityto the DSS has been positive. In the same context, Vinnikand Scholl [24] proposed UNICAP (acronym for university’scapacity planning), which was aimed at optimizing theacademic decision-making and admission capacity planning,by allowing simulation and evaluation of strategic plans. Thesystem integrates data from heterogeneous sources, appliesOLAP (online analytical processing) and data warehousetechniques, and allows users to interact with it in order totest various development strategies and become aware of theirquantitative implications. The user interface of UNICAP isassured by providing orientation aids, detailed instructions,and graphical support and leading the user through thecomputation. Visually enhanced presentation of the outputfacilitates its perception and interpretation. Recently, mul-tiaggregator models for fuzzy queries and ranking basedon an evolutionary computing approach to build a decisionsupport system for admission student in university have beenintroduced by Alsharafat [25]. A unified approach based ona combination of four soft computing methodologies (FuzzyLogic, Neuronetworks, GeneticAlgorithms, andProbabilisticReasoning) was used to build the proposed intelligent DSS.The information provided in this study was a hypotheticalsituation that will reflect future admissions criteria. Based onthe above literature review from different points of view todeal with this problem through years, it can be concludedthat the resource planning of higher education in universityis still an open research area and many further studies mustbe carried out in different directions to build an efficient,effective, and integrated decision support system able to solvethemajority of subsystems of university resource planning byincorporating simulation-optimization computer programsand intelligent data processing techniques to answer what-if scenarios and determining the optimal one or, alterna-tively, develop an appropriate optimization methodologyusing mathematical methods coupling with soft computingtechniques to handle one subsystem proficiently by takinginto consideration all factors that affect it as presented in theproposed research work.
3. System General Outlines
As can be clearly seen from the previously studied literaturereview, a considerable number of research studies have beenproposed to model university admission planning problems.Some of these studies used goal programming technique
4 Computational Intelligence and Neuroscience
Control education tracks
Satisfy job market
requirements
Fair student satisfaction
Model objectives
LevelTrack
Place
Budget
Facilities
Faculty
Mathematical model
Social needs
Higher education
plan
Educationquality
Inputs
Data base
Resources
Government
Students
Decision
variables
Goals
Constraints
Objective
function
Satisfy resource constraints
Figure 1: Admission system general outlines view.
alone or combined with other classical mathematical meth-ods or intelligent optimization algorithms.
The scope of all of these mentioned studies was limitedto the admission planning related to one unique collegeor institution. Most of them are concerned with availablecapacity and resources. Additionally, the planning horizonwas limited to one upcoming year. The tradition way usedfor tackling such a problem is to design a separate plan thatalignswith the specific needs and behaviors of each individualinstitute to cope with the available resources and capacity.None of the mentioned studies tackled a global view andsolve the comprehensive problem at a national level and solvefor accomplishing the national objectives of a country withrespect to admission problem in higher education takinginto consideration the social needs and the various objectivesstated in the national development plan.
The current paper presents a new more generalizedmathematical model for higher education sector in two ways:application on the national level and the long-term planninghorizon. A case study in the Kingdom of Saudi Arabia ispresented to clarify the idea. Meanwhile, the proposedmodelis designed in a general way that it can be adopted to be suitedfor different countries and/or universities.
Similar to higher education systems throughout much ofthe world higher education, Saudi Arabia faces a numberof challenges. Among such challenges are the social needsexpressed by the increasing demand for higher educationand the correspondence between the admission capacity andthe population, the available resources expressed by facilities,faculty, and budget, student needs expressed by location andeducation track limited by student levels, and the job marketneeds. Figure 1 demonstrates all the mentioned factors asinputs to the proposed admission system outlines together
with the components of the mathematical model and itsobjectives.
The proposed goal programming model will consider allthe above inputs; then it will design the required unknowndecision variables, goals, constraints, and objective functionto satisfy the long-term plan that would help face up thevarious challenges standing in the way of all the highereducational institutions.
The main objectives for the model are to cope withthe increasing demand for higher education in the countryand to satisfy the job market requirements, fair studentsatisfaction, and control over the education tracks (medical,science and engineering, and arts) under the limitation ofavailable resources.
4. Solution Methodology
The algorithm of solution starts with studying the strategicplans in the country related to higher education to extractthe objectives intended for the admission problem. Figure 2represents the complete steps to solve the problem fordifferent years of the planning horizon, 𝑛. Since each yearhas its specific data, the mode will be formulated and solvedinitially for the first year. The obtained results will be givenas some of the input data for the second year, and so on, tillreaching the last year of the plan.
5. Goal Programming Model forthe Admission Problem
Firms often have more than one goal; they may want toachieve several, sometimes contradictory, goals. It is not
Computational Intelligence and Neuroscience 5
Data collection for the
Formulate the relevant goalprogramming model
Solve the mathematical model
Final conclusions andrecommendations
Start
Design the different attributesfor the admission problem
Study input parameters andmodel objectives
for the admission problem
End
Results aresatisfactory?
Planning year y = 1
y = y + 1
Year y = n is reached?
planning year y
Filter input parameters and objectives
Figure 2: Flowchart of the general solution methodology.
always possible to satisfy every goal so goal programmingattempts to reach a satisfactory level of multiple objectives.
The main difference is in the objective function wheregoal programming tries to minimize the deviations betweengoals and what we can actually achieve within the givenconstraints. The mathematical model will cover the mainobjectives stated in the current KSA Development Plan andthat stated in KSAHigher Education Strategic Plan (AAFAQ)for the next 25 years. It will be restricted also to the budget andstaff constraints as problem resources.
5.1. Decision Variables. To design the decision variables forthe problem, it is necessary to represent all the differentproblem attributes. Let the decision variables be denoted by𝑥𝑦,𝑢
𝑖,𝑗,𝑘= 𝑥
year of the plan,universitystatus, gender, program = number of students, where
different attributes are shown in Table 1.
5.2. Problem Goals. The Kingdom Development Plan andAAFAQ Project objectives are formulated to represent themathematical model goals as follows.
Let
𝑑𝑦−
𝑛be underachievement of the 𝑛th target in year 𝑦,
𝑑𝑦+
𝑛be overachievement of the 𝑛th target in year 𝑦,
where 𝑛 is the number of the constraints.
Table 1: Problem attributes and their values.
Attributes Values
𝑦 = year ofthe plan
𝑦 = 1 for the first year of the next plan (2015), 2 forthe second, 3 for the third, 4 for the fourth, and 5for the last𝑦 = 0 for the last year in the previousplan (current year, 2014), −1 for 2013, and so on
𝑢 = university 𝑢 = a university, 𝑢 ∈ 𝑈, the set of all universities inthe country
𝑖 = status 𝐸 stands for “enrolled” and 𝐺 for “graduated”𝑗 = gender 𝑏 stands for boys section and 𝑔 for girls section
𝑘 = educationprogram
𝑚 = a college in the medicine specialty,𝑚 ∈ 𝑀,the set of all colleges in the medicine specialty𝑠 = a college in the science and engineeringspecialty, 𝑠 ∈ 𝑆, the set of all colleges in the scienceand engineering specialty𝑎 = a college in the arts specialty, 𝑎 ∈ 𝐴, the set ofall colleges in the arts specialty, and 𝑇 is the totalnumber in all specialties𝑀, 𝑆, and 𝐴
The deviation variables that need to be minimized willbe included in the objective function and in the corre-sponding constraint, but those whose values are permittedto have nonzero positive values will be omitted from boththe objective function and the corresponding constraints.
6 Computational Intelligence and Neuroscience
The constraints are adjusted correspondingly to be of equalityor nonequality types.
5.2.1. Increase in the Enrollment Rate. It is required to increasethe enrollment of students in higher education with anaverage annual growth rate of 𝑝𝑦
𝑟:
∑𝑢∈𝑈
∑𝑘∈𝐾
𝑥𝑦,𝑢
𝐸,𝑏,𝑘− ∑𝑢∈𝑈
∑𝑘∈𝐾
𝑥𝑦−1,𝑢
𝐸,𝑏,𝑘
∑𝑢∈𝑈
∑𝑘∈𝐾
𝑥𝑦−1,𝑢
𝐸,𝑏,𝑘
+ 𝑑𝑦−
1≥ 𝑝𝑦
𝑟
∑𝑢∈𝑈
∑𝑘∈𝐾
𝑥𝑦,𝑢
𝐸,𝑔,𝑘− ∑𝑢∈𝑈
∑𝑘∈𝐾
𝑥𝑦−1,𝑢
𝐸,𝑔,𝑘
∑𝑢∈𝑈
∑𝑘∈𝐾
𝑥𝑦−1,𝑢
𝐸,𝑔,𝑘
+ 𝑑𝑦−
2≥ 𝑝𝑦
𝑟
𝑦 = 1, 2, 3, 4, 5.
(1)
5.2.2. Control the Education Tracks. The percentage of thetotal number of students enrolled in science and technologyprograms to the total number of students enrolled in highereducation = 𝑝
𝑦
𝑠and the percentage of the total number of
students enrolled in medical programs to the total numberof students enrolled in science and engineering = 𝑝
𝑦
𝑚are as
follows:
∑𝑢∈𝑈
∑𝑘=𝑚,𝑠
𝑥𝑦,𝑢
𝐸,𝑏,𝑘
∑𝑢∈𝑈
∑𝑗∈𝐽
𝑥𝑦−1,𝑢
𝐸,𝑏,𝑗
+ 𝑑𝑦−
3− 𝑑𝑦+
3= 𝑝𝑦
𝑠,
∑𝑢∈𝑈
∑𝑘=𝑚,𝑠
𝑥𝑦,𝑢
𝐸,𝑔,𝑘
∑𝑢∈𝑈
∑𝑘∈𝐾
𝑥𝑦−1,𝑢
𝐸,𝑔,𝑘
+ 𝑑𝑦−
4− 𝑑𝑦+
4= 𝑝𝑦
𝑠,
∑𝑢∈𝑈
∑𝑘=𝑚∈𝑀
𝑥𝑦,𝑢
𝐸,𝑏,𝑘
∑𝑢∈𝑈
∑𝑘=𝑠∈𝑆
𝑥𝑦−1,𝑢
𝐸,𝑏,𝑗
+ 𝑑𝑦−
5− 𝑑𝑦+
5= 𝑝𝑦
𝑚,
∑𝑢∈𝑈
∑𝑘=𝑚∈𝑀
𝑥𝑦,𝑢
𝐸,𝑔,𝑘
∑𝑢∈𝑈
∑𝑘=𝑠∈𝑆
𝑥𝑦−1,𝑢
𝐸,𝑔,𝑘
+ 𝑑𝑦−
6− 𝑑𝑦+
6= 𝑝𝑦
𝑚,
𝑦 = 1, 2, 3, 4, 5.
(2)
5.2.3. Percentage of Enrollment to Population and Percentageof Enrollment to High School Graduates. The percentage oftotal enrollment in higher education, regardless of age, to thetotal population in the age group of 18–23 years ≥ 𝑝
𝑦
𝑝 andthe accepted percentage in higher education fromhigh schoolgraduates in the same year, 𝑝𝑦
ℎ, are as follows:
∑
𝑢∈𝑈
∑
𝑗∈𝐽
∑
𝑘∈𝐾
𝑥𝑦,𝑢
𝐸,𝑗,𝑘+ 𝑑𝑦−
7
≥ max [(1
5
⋅ 𝑝𝑦
𝑝⋅ 𝑁𝑦−1
𝐶) , (𝑝𝑦
ℎ⋅ 𝑁𝑦−1
𝐻)] ,
𝑦 = 1, 2, 3, 4, 5,
(3)
where 𝑝𝑦
𝑝 is the percentage of total enrollment in highereducation, regardless of age, to the total population in theage group of 18–23 years in year 𝑦. 𝑁𝑦
𝐶is the population of
Saudi Arabia in the age of 18–23 years in year 𝑦. 𝑝𝑦ℎis the
accepted percentage in higher education from high schoolgraduates in year 𝑦. 𝑁𝑦
𝐻represents high school graduates
in year 𝑦 that will be decreased by the number of boys forbachelor scholarships abroad (𝑁
𝑦−1
𝑏) and the number of girls
for bachelor scholarships abroad (𝑁𝑦−1
𝑔).
5.2.4. Enrollment and Student-to-Faculty Ratio. The percent-ages of the total number of students in each discipline ofuniversity education to the total faculty (𝐹) in that specialtyare as follows:
medicine (𝑚 ∈ 𝑀) = 𝛽𝑦
𝑀,
engineering and science (𝑒 ∈ 𝐸) = 𝛽𝑦
𝐸,
arts (𝑎 ∈ 𝐴) = 𝛽𝑦
𝐴,
total university (𝑢 ∈ 𝑈) = 𝛽𝑦
𝑈:
∑
𝑢∈𝑈
∑
𝑚∈𝑀
𝑥𝑦,𝑢
𝐸,𝑏,𝑚− 𝑑𝑦+
8≤
1
𝑡𝑀
⋅ 𝛽𝑦
𝑀⋅ 𝐹𝑦,𝑈
𝑏,𝑀,
∑
𝑢∈𝑈
∑
𝑚∈𝑀
𝑥𝑦,𝑢
𝐸,𝑔,𝑚− 𝑑𝑦+
9≤
1
𝑡𝑀
⋅ 𝛽𝑦
𝑀⋅ 𝐹𝑦,𝑈
𝑔,𝑀,
∑
𝑢∈𝑈
∑
𝑠∈𝑆
𝑥𝑦,𝑢
𝐸,𝑏,𝑠− 𝑑𝑦+
10≤
1
𝑡𝑆
⋅ 𝛽𝑦
𝑆⋅ 𝐹𝑦,𝑈
𝑏,𝑆,
∑
𝑢∈𝑈
∑
𝑠∈𝑆
𝑥𝑦,𝑢
𝐸,𝑔,𝑠− 𝑑𝑦+
11≤
1
𝑡𝑆
⋅ 𝛽𝑦
𝑆⋅ 𝐹𝑦,𝑈
𝑔,𝑀,
∑
𝑢∈𝑈
∑
𝑎∈𝐴
𝑥𝑦,𝑢
𝐸,𝑏,𝑎− 𝑑𝑦+
12≤
1
𝑡𝐴
⋅ 𝛽𝑦
𝐴⋅ 𝐹𝑦,𝑈
𝑏,𝐴,
∑
𝑢∈𝑈
∑
𝑎∈𝐴
𝑥𝑦,𝑢
𝐸,𝑔,𝑎− 𝑑𝑦+
13≤
1
𝑡𝐴
⋅ 𝛽𝑦
𝐴⋅ 𝐹𝑦,𝑈
𝑔,𝐴,
∑
𝑢∈𝑈
∑
𝑘∈𝐾
𝑥𝑦,𝑢
𝐸,𝑏,𝑘− 𝑑𝑦+
14≤
1
𝑡𝑀
⋅ 𝛽𝑦
𝑈⋅ 𝐹𝑦,𝑈
𝑏,𝑇,
∑
𝑢∈𝑈
∑
𝑘∈𝐾
𝑥𝑦,𝑢
𝐸,𝑔,𝑘− 𝑑𝑦+
15≤
1
𝑡𝑀
⋅ 𝛽𝑦
𝑈⋅ 𝐹𝑦,𝑈
𝑔,𝑇,
𝑦 = 1, 2, 3, 4, 5,
(4)
where 𝐹𝑦,𝑈𝑗,𝐾
is the number of faculty members for gender 𝑗 inspecialty 𝐾 in all universities 𝑈; 𝑗 = 𝑏, 𝑔;𝐾 = 𝑀, 𝑆, 𝐴 and 𝑈;𝑡𝑘 is the number of years in a program 𝑘.
5.2.5. Resources Constraints for Enrollment. All the resourcesof the teaching process are collected in the total budgetrequired for a university 𝑢 that should not exceed a certaintotal limit of 𝐵𝑦
𝑢at any year 𝑦 of the planning horizon.
Let
𝑐𝑦,𝑢
𝑢be the cost per student in a university 𝑢 in a year
𝑦,𝐵𝑦
𝑢be themaximumbudget for a university 𝑢 in a year
𝑦, 𝑦 = 1, 2, 3, 4, 5.
Computational Intelligence and Neuroscience 7
Then,
∑
𝑢∈𝑈
∑
𝑗∈𝐽
∑
𝑘∈𝐾
𝑥𝑦,𝑢
𝐸,𝑗,𝑘− 𝑑𝑦+
16≤ ∑
𝑢∈𝑈
𝐵𝑦
𝑢
𝑐𝑦,𝑢
𝑢
, 𝑦 = 1, 2, 3, 4, 5. (5)
5.2.6. Student Fair Satisfaction. The student fair satisfactionwith respect to the geographical location for enrollment inthe nearest university to his/her town and the educationtrack he/she prefers will be fairly accomplished for all thestudents according to the available places and the preferenceparameters such as GPA, home town and marks in differentcourses, and the student’s prioritized desires.
5.2.7. Logic Constraints. The total number of students inall the universities in any year 𝑦 is equal to the totalnumber of students in all the specialties: medical, science andengineering, and arts; this is applied for both boys and girlsas follows:
∑
𝑢∈𝑈
∑
𝑘∈𝐾
𝑥𝑦,𝑢
𝐸,𝑏,𝑘= ∑
𝑢∈𝑈
∑
𝑘=𝑚
𝑥𝑦,𝑢
𝐸,𝑏,𝑘+ ∑
𝑢∈𝑈
∑
𝑘=𝑠
𝑥𝑦,𝑢
𝐸,𝑏,𝑘
+ ∑
𝑢∈𝑈
∑
𝑘=𝑎
𝑥𝑦,𝑢
𝐸,𝑏,𝑘,
∑
𝑢∈𝑈
∑
𝑘∈𝐾
𝑥𝑦,𝑢
𝐸,𝑔,𝑘= ∑
𝑢∈𝑈
∑
𝑘=𝑚
𝑥𝑦,𝑢
𝐸,𝑔,𝑘+ ∑
𝑢∈𝑈
∑
𝑘=𝑠
𝑥𝑦,𝑢
𝐸,𝑔,𝑘
+ ∑
𝑢∈𝑈
∑
𝑘=𝑎
𝑥𝑦,𝑢
𝐸,𝑔,𝑘
𝑦 = 1, 2, 3, 4, 5.
(6)
5.2.8. Increase in the Graduated Students. With the planprojects over the same period, the number of graduates willincrease with an average annual rate of 𝑞𝑦
𝑟.
5.2.9. Increase in the Graduation Rate. The percentage ofstudents who have completed their studies in a given year tothe total number of students enrolled in universities five yearsbefore that year is 𝑞𝑦
𝑑.
These two goals will be expressed as follows:
∑
𝑢∈𝑈
∑
𝑚∈𝑀
𝑥𝑦,𝑢
𝐺,𝑏,𝑚+ 𝑑𝑦−
17≥ max[(1 + 𝑞
𝑦
𝑟)
⋅ ∑
𝑢∈𝑈
∑
𝑚∈𝑀
𝑥𝑦−1,𝑢
𝐺,𝑏,𝑚, 𝑞𝑦
𝑑⋅ ∑
𝑢∈𝑈
∑
𝑚∈𝑀
𝑥𝑦−5,𝑢
𝐸,𝑏,𝑚] ,
∑
𝑢∈𝑈
∑
𝑠∈𝑆
𝑥𝑦,𝑢
𝐺,𝑏,𝑠+ 𝑑𝑦−
18≥ max[(1 + 𝑞
𝑦
𝑟) ⋅ ∑
𝑢∈𝑈
∑
𝑠∈𝑆
𝑥𝑦−1,𝑢
𝐺,𝑏,𝑠, 𝑞𝑦
𝑑
⋅ ∑
𝑢∈𝑈
∑
𝑠∈𝑆
𝑥𝑦−5,𝑢
𝐸,𝑏,𝑠] ,
∑
𝑢∈𝑈
∑
𝑎∈𝐴
𝑥𝑦,𝑢
𝐺,𝑏,𝑎+ 𝑑𝑦−
19≥ max[(1 + 𝑞
𝑦
𝑟) ⋅ ∑
𝑢∈𝑈
∑
𝑎∈𝐴
𝑥𝑦−1,𝑢
𝐺,𝑏,𝑎, 𝑞𝑦
𝑑
⋅ ∑
𝑢∈𝑈
∑
𝑎∈𝐴
𝑥𝑦−5,𝑢
𝐸,𝑏,𝑎] ,
∑
𝑢∈𝑈
∑
𝑚∈𝑀
𝑥𝑦,𝑢
𝐺,𝑔,𝑚+ 𝑑𝑦−
20≥ max[(1 + 𝑞
𝑦
𝑟)
⋅ ∑
𝑢∈𝑈
∑
𝑚∈𝑀
𝑥𝑦−1,𝑢
𝐺,𝑔,𝑚, 𝑞𝑦
𝑑⋅ ∑
𝑢∈𝑈
∑
𝑚∈𝑀
𝑥𝑦−5,𝑢
𝐸,𝑔,𝑚] ,
∑
𝑢∈𝑈
∑
𝑠∈𝑆
𝑥𝑦,𝑢
𝐺,𝑔,𝑠+ 𝑑𝑦−
21≥ max[(1 + 𝑞
𝑦
𝑟) ⋅ ∑
𝑢∈𝑈
∑
𝑠∈𝑆
𝑥𝑦−1,𝑢
𝐺,𝑔,𝑠, 𝑞𝑦
𝑑
⋅ ∑
𝑢∈𝑈
∑
𝑠∈𝑆
𝑥𝑦−5,𝑢
𝐸,𝑔,𝑠] ,
∑
𝑢∈𝑈
∑
𝑎∈𝐴
𝑥𝑦,𝑢
𝐺,𝑔,𝑎+ 𝑑𝑦−
22≥ max[(1 + 𝑞
𝑦
𝑟) ⋅ ∑
𝑢∈𝑈
∑
𝑎∈𝐴
𝑥𝑦−1,𝑢
𝐺,𝑔,𝑎, 𝑞𝑦
𝑑
⋅ ∑
𝑢∈𝑈
∑
𝑎∈𝐴
𝑥𝑦−5,𝑢
𝐸,𝑔,𝑎]
𝑦 = 1, 2, 3, 4, 5.
(7)
5.2.10. Job Market Requirements. Let 𝑁𝑦𝑗,𝑘
be the number ofavailable jobs in the country for gender 𝑗 = 𝑏 and 𝑔 andspecialty 𝑘 = 𝑚, 𝑠, and 𝑒, in a year 𝑦; then,
∑
𝑢∈𝑈
∑
𝑚∈𝑀
𝑥𝑦,𝑢
𝐺,𝑏,𝑚− 𝑑𝑦+
23≤ 𝑁𝑦
𝑏,𝑚,
∑
𝑢∈𝑈
∑
𝑠∈𝑆
𝑥𝑦,𝑢
𝐺,𝑏,𝑠− 𝑑𝑦+
24≤ 𝑁𝑦
𝑏,𝑠,
∑
𝑢∈𝑈
∑
𝑎∈𝐴
𝑥𝑦,𝑢
𝐺,𝑏,𝑎− 𝑑𝑦+
25≤ 𝑁𝑦
𝑏,𝑎,
∑
𝑢∈𝑈
∑
𝑚∈𝑀
𝑥𝑦,𝑢
𝐺,𝑔,𝑚− 𝑑𝑦+
26≤ 𝑁𝑦
𝑔,𝑚,
∑
𝑢∈𝑈
∑
𝑠∈𝑆
𝑥𝑦,𝑢
𝐺,𝑔,𝑠− 𝑑𝑦+
27≤ 𝑁𝑦
𝑔,𝑠,
∑
𝑢∈𝑈
∑
𝑎∈𝐴
𝑥𝑦,𝑢
𝐺,𝑔,𝑎− 𝑑𝑦+
28≤ 𝑁𝑦
𝑔,𝑎,
𝑦 = 1, 2, 3, 4, 5.
(8)
5.3. Objective Function. Once all goals and constraints areidentified, management should analyze each goal to seewhether underachievement or overachievement of that goalis an acceptable situation:
(i) If overachievement is acceptable, the appropriatecorresponding deviation variable can be eliminatedfrom the objective function.
(ii) If underachievement is okay, the corresponding devi-ation variable should be dropped.
(iii) If management seeks to attain a goal exactly, bothdeviation variables must appear in the objective func-tion.
8 Computational Intelligence and Neuroscience
Typically, goals set by management can be achieved only atthe expense of other goals. A hierarchy of importance needsto be established so that higher-priority goals are satisfiedbefore lower-priority goals are addressed. Priorities (𝑃𝑖’s) areassigned to each deviational variable with the ranking so that𝑃1 is the most important goal, 𝑃2 the next most important, 𝑃3the third, and so on.
In our problem formulation, the goals and systems relatedto planning year 1 have the highest priority, and those relatedto planning year 2 are higher than those related to years 3,4, and 5, and so on. The same weights will be given to allthe goals in the same priority level. Accordingly, the problemwill be divided into several problems; each one is related onlyto one planning year. The results obtained from each prioritywill be considered as constraints for the other priority levels.
So, the objective function is formulated as follows:
Minimize: 𝑧 = 𝑑𝑦−
1+ 𝑑𝑦−
2+ 𝑑𝑦−
3+ 𝑑𝑦+
3+ 𝑑𝑦−
4+ 𝑑𝑦+
4
+ 𝑑𝑦−
5+ 𝑑𝑦+
5+ 𝑑𝑦−
6+ 𝑑𝑦+
6+ 𝑑𝑦−
7+ 𝑑𝑦+
8
+ 𝑑𝑦+
9+ 𝑑𝑦+
10+ 𝑑𝑦+
11+ 𝑑𝑦+
12+ 𝑑𝑦+
13+ 𝑑𝑦+
14
+ 𝑑𝑦+
15+ 𝑑𝑦+
16+ 𝑑𝑦−
17+ 𝑑𝑦−
18+ 𝑑𝑦−
19+ 𝑑𝑦−
20
+ 𝑑𝑦−
21+ 𝑑𝑦−
22+ 𝑑𝑦+
23+ 𝑑𝑦+
24+ 𝑑𝑦+
25+ 𝑑𝑦+
26
+ 𝑑𝑦+
27+ 𝑑𝑦+
28.
(9)
6. Kingdom of Saudi Arabia as a Case Study
Substituting 𝑦 = 1 for the whole mathematical model, wewill have the first priority level and the first part of themathematical model. The following data are collected relatedto the Kingdom of Saudi Arabia; see Table 2.
It can be noticed that the mathematical model containstwo distinct parts; one is related to the enrollment processwhile the other is related to the graduation process. Theenrollment part is common for the first 18 goals, while thegraduation part is specified in the remaining constraintswhile their related decision variables are not included in boththe objective function and the enrollment constraints.
Constraints numbers 19–24 concerning the number ofgraduates can be completely satisfied without any effect onother parts of the enrollment process.
The problem of student enrollment is solved using theproposed differential evolution approachwith data represent-ing the first year of the National Plan for the Kingdom ofSaudi Arabia.
7. The Proposed DifferentialEvolution Approach
It can be seen that the proposed GPM contains nonlinearconstraints and involves a large amount of integer variablesand it is not as simple as the linear GPmodel with continuousvariables. Therefore, a novel constrained optimization basedonmodified differential evolution algorithmnamedCOMDE(Mohamed and Sabry, 2012) [26] is used to solve the proposed
nonlinear integer GP problem. Actually, the effectivenessand benefits of the new directed mutation strategy andmodified basic strategy used in COMDE have been exper-imentally investigated. Numerical experiments on 13 well-known benchmark test functions and five engineering designproblems have shown that the new approach is efficient,effective, and robust. The comparison results between theCOMDE and the other twenty-eight state-of-the-art evo-lutionary algorithms indicate that the proposed COMDEalgorithm is competitive with, and in some cases superior to,other existing algorithms in terms of the quality, efficiency,convergence rate, and robustness of the final solution. Thus,due its previous success, it is used here as an optimizationapproach with simple modification to handle integer vari-ables, as will be mentioned in Section 7.3, without any mod-ification to solve admission problems in higher education.Consequently, we use our algorithm to solve a real worldproblem which is similar to other benchmark problems intheir mathematical features. Differential evolution (DE) hasbeen receiving great attention and has also been successfullyapplied in many research fields in the last decade (Das andSuganthan, 2011) [27]. However, to the best of our knowledge,this is the first time to use DE in solving admission problemsin higher education.
7.1. Differential Evolution (DE). Differential evolution (DE)is a stochastic population-based search method, proposedby Das and Suganthan [27]. DE is relatively recent EAs forsolving real-parameter optimization problems [28]. DE hasmany advantages including simplicity of implementation,reliability, and robustness and in general is considered as aneffective global optimization algorithm [29]. In this paper,the scheme which can be classified using the notation asDE/rand/1/bin strategy is used [30, 31]. This strategy isthe most often used in practice. A set of 𝐷 optimizationparameters is called an individual, which is represented by a𝐷-dimensional parameter vector.
A population consists of NP parameter vectors 𝑥𝐺
𝑖, 𝑖 =
1, 2, . . . ,NP. 𝐺 denotes one generation.NP is the number of members in a population. It does not
change during the process. The initial population is chosenrandomly with uniform distribution in the search space.DE has three operators: mutation, crossover, and selection.The crucial idea behind DE is a scheme for generatingtrial vectors. Mutation and crossover operators are usedto generate trial vectors, and the selection operator thendetermines which of the vectors will survive into the nextgeneration [31–34].
7.1.1. Initialization. In order to establish a starting pointfor the optimization process, an initial population must becreated. Typically, each decision parameter in every vectorof the initial population is assigned a randomly chosen valuefrom the boundary constraints:
𝑥0
𝑖𝑗= 𝑙𝑗 + rand𝑗 ∗ (𝑢𝑗 − 𝑙𝑗) , (10)
where rand𝑗 denotes a uniformly distributed number inthe range [0, 1], generating a new value for each decision
Computational Intelligence and Neuroscience 9
Table 2: Relevant data related to the Kingdom of Saudi Arabia.
Sym. Meaning Value
𝑥0,𝑈
𝐸,𝑏,𝑇 Total number of boys enrolled in Saudi Arabia in all universities in 2014 (last year) 205,362𝑥0,𝑈
𝐸,𝑔,𝑇 Total number of girls enrolled in Saudi Arabia in all universities in 2014 (last year) 190,608𝑁0
𝐶 Population of Saudi Arabia in the age of 18–23 years in 2014 1,036,700𝑁0
𝐻 High school graduates in the kingdom in 2014 380,050𝑁0
𝑏 Number of boys for bachelor scholarships abroad in 2014 22,644𝑁0
𝑔 Number of girls for bachelor scholarships abroad in 2014 8,477𝐹1,𝑈
𝑏,𝑀 Current number of faculties in all universities (boys section, medical specialty) 7,425𝐹1,𝑈
𝑔,𝑀 Current number of faculties in all universities (girls section, medical specialty) 4,433𝐹1,𝑈
𝑏,𝑆 Current number of faculties in all universities (boys section, science and engineering specialty) 19,346𝐹1,𝑈
𝑔,𝑆 Current number of faculties in all universities (girls section, science and engineering specialty) 8,658𝐹1,𝑈
𝑏,𝐴 Current number of faculties in all universities (boys section, arts specialty) 9,431𝐹1,𝑈
𝑔,𝐴 Current number of faculties in all universities (girls section, arts specialty) 9,776𝐹1,𝑈
𝑏,𝑇 Current number of faculties in all universities (boys section, all specialties) 37,245𝐹1,𝑈
𝑔,𝑇 Current number of faculties in all universities (girls section, all specialties) 23,405𝐵1,𝑈 Total budget for the current year in all universities in the kingdom 4,778.5 million
𝑐1,𝑈 Average cost of one student at the kingdom level in the current year 56,250𝑥−5,𝑈
𝐸,𝑏,𝑀 Number of enrolled boys in the kingdom in 2009 (medicine) 8,518𝑥0,𝑈
𝐺,𝑏,𝑀 Number of graduated boys in the kingdom in 2014 (medicine) 3,191𝑥−4,𝑈
𝐸,𝑏,𝑆 Number of enrolled boys in the kingdom in 2010 (science and engineering) 93,807𝑥0,𝑈
𝐺,𝑏,𝑆 Number of graduated boys in the kingdom in 2014 (science and engineering) 18,103𝑥−3,𝑈
𝐸,𝑏,𝐴 Number of enrolled boys in the kingdom in 2011 (arts) 34,301𝑥0,𝑈
𝐺,𝑏,𝐴 Number of graduated boys in the kingdom in 2014 (arts) 13,851𝑥−5,𝑈
𝐸,𝑔,𝑀 Number of enrolled girls in the kingdom in 2009 (medicine) 7,114𝑥0,𝑈
𝐺,𝑔,𝑀 Number of graduated girls in the kingdom in 2014 (medicine) 3,456𝑥−4,𝑈
𝐸,𝑔,𝑆 Number of enrolled girls in the kingdom in 2010 (science and engineering) 92,867𝑥0,𝑈
𝐺,𝑔,𝑆 Number of graduated girls in the kingdom in 2014 (science and engineering) 22,871𝑥−3,𝑈
𝐸,𝑔,𝐴 Number of enrolled girls in the kingdom in 2011 (arts) 38,993𝑥0,𝑈
𝐺,𝑔,𝐴 Number of graduated girls in the kingdom in 2014 (arts) 34,797
parameter. 𝑙𝑖 and 𝑢𝑖 are the lower and upper bounds for the𝑗th decision parameter, respectively [28].
7.1.2. Mutation. For each target vector 𝑥𝐺𝑖, a mutant vector V
is generated according to the following:
V𝐺+1𝑖
= 𝑥𝐺
𝑟1
+ 𝐹 ∗ (𝑥𝐺
𝑟2
− 𝑥𝐺
𝑟3
) , 𝑟1 = 𝑟2 = 𝑟3 = 𝑖, (11)
with randomly chosen indices and 𝑟1, 𝑟2, and 𝑟3 ∈
{1, 2, . . . ,NP}.Note that these indices have to be different from each
other and from the running index 𝑖 so that NP must be atleast four. 𝐹 is a real number to control the amplification ofthe difference vector (𝑥𝐺
𝑟2
− 𝑥𝐺
𝑟3
). According to [29], the rangeof 𝐹 is in [0, 2]. If a component of a mutant vector goes offthe search space, that is, if a component of a mutant vectorviolates the boundary constraints, then the new value of thiscomponent is generated using (10).
7.1.3. Crossover. The target vector is mixed with the mutatedvector, using the following scheme, to yield the trial vector 𝑢:
𝑢𝐺+1
𝑖𝑗=
{
{
{
V𝐺+1𝑖𝑗
, rand (𝑗) ≤ CR or 𝑗 = rand 𝑛 (𝑖) ,
𝑥𝐺
𝑖𝑗, rand (𝑗) > CR and 𝑗 = rand 𝑛 (𝑖) ,
(12)
where 𝑗 = 1, 2, . . . , 𝐷 and rand(𝑗) ∈ [0, 1] is the 𝑗th evaluationof a uniform random generator number. CR ∈ [0, 1] is thecrossover probability constant, which has to be determined bythe user. rand𝑛(𝑖) ∈ {1, 2, . . . , 𝐷} is a randomly chosen indexwhich ensures that 𝑢𝐺+1
𝑖gets at least one element from V𝐺+1
𝑖.
7.1.4. Selection. DE adapts a greedy selection strategy. If andonly if the trial vector 𝑢
𝐺+1
𝑖yields a better fitness function
value than 𝑥𝐺
𝑖, then 𝑢
𝐺+1
𝑖is set to 𝑥
𝐺+1
𝑖. Otherwise, the old
10 Computational Intelligence and Neuroscience
(1) Begin(2) 𝐺 = 0
(3) Create a random initial population ��𝐺
𝑖∀𝑖, 𝑖 = 1, . . . ,NP
(4) Evaluate 𝑓(��𝐺𝑖) ∀𝑖, 𝑖 = 1, . . . ,NP
(5) For 𝐺 = 1 to GEN Do(6) For 𝑖 = 1 to NP Do(7) Select randomly 𝑟1 = 𝑟2 = 𝑟3 = 𝑖 ∈ [1,NP](8) 𝑗rand = randint(1, 𝐷)
(9) For 𝑗 = 1 to𝐷 Do(10) If (rand
𝑗[0, 1] < CR or 𝑗 = 𝑗rand) Then
(11) 𝑢𝐺+1
𝑖,𝑗= 𝑥𝐺
𝑟1,𝑗+ 𝐹 ⋅ (𝑥
𝐺
𝑟2,𝑗− 𝑥𝐺
𝑟3,𝑗)
(12) Else(13) 𝑢
𝐺+1
𝑖,𝑗= 𝑥𝐺
𝑖,𝑗
(14) End If(15) End For(16) Verify Boundary constraints(16) If (𝑓(��𝐺+1
𝑖) ≤ 𝑓(��
𝐺
𝑖)) Then
(17) ��𝐺+1
𝑖= ��𝐺+1
𝑖
(18) Else(19) ��
𝐺+1
𝑖= ��𝐺
𝑖
(20) End If(21) End For(22) 𝐺 = 𝐺 + 1
(23) End For(24) End
Algorithm 1: Description of standard DE algorithm.
value 𝑥𝐺𝑖is retained. The selection scheme is as follows (for a
minimization problem):
𝑥𝐺+1
𝑖=
{
{
{
𝑢𝐺+1
𝑖, 𝑓 (𝑢
𝐺+1
𝑖) < 𝑓 (𝑥
𝐺
𝑖) ,
𝑥𝐺
𝑖, 𝑓 (𝑢
𝐺+1
𝑖) ≥ 𝑓 (𝑥
𝐺
𝑖) .
(13)
A detailed description of standard DE algorithm is given inAlgorithm 1.
rand[0, 1) is a function that returns a real numberbetween 0 and 1. randint (min,max) is a function that returnsan integer number between min and max. NP, GEN, CR, and𝐹 are user-defined parameters.𝐷 is the dimensionality of theproblem.
7.2. Constrained Optimization Based on Modified DifferentialEvolution Algorithm (COMDE). All evolutionary algorithms,including DE, are stochastic population-based search meth-ods. Accordingly, there is no guarantee that the global optimalsolution will be reached consistently. Furthermore, they arenot originally designed to solve constrained optimizationproblems. Nonetheless, adjusting control parameters suchas the scaling factor, the crossover rate, and the populationsize, alongside developing an appropriate mutation schemeand coupling with suitable and effective constraint-handlingtechniques, can considerably improve the search capabilityof DE algorithms. Therefore, in the proposed algorithm, anew directedmutation rule, based on the weighted differencevector between the best and the worst individuals at a par-ticular generation, is introduced. The new directed mutation
rule is combined with the modified basic mutation strategyDE/rand/1/bin, where only one of the two mutation rules isapplied with the probability of 0.5. The proposed mutationrule is shown to enhance the local search ability of thebasic differential evolution (DE) and to get a better tradeoffbetween convergence rate and robustness.
Two new scaling factors are introduced as uniformrandom variables to improve the diversity of the populationand to bias the search direction. Additionally, a dynamicnonlinear increased crossover probability is utilized to bal-ance the global exploration and local exploitation. COMDEalso includes amodified constraint-handling technique basedon feasibility and the sum of constraints violations. A newdynamic tolerance technique to handle equality constraintsis also adopted. However, the test problem contains manyequality constraints which is considered a very difficultproblem. Thus, in order to increase the number of infeasiblesolutions to be improved through generations and becomefeasible with true feasible region, the initial tolerance 𝑎 = 100,where 𝐹initial = −log10(𝑎), 𝐹final = 4, and 𝑅 = 0.75, thefactor equation decreases linearly with 𝑘 = 1. The requiredpopulation size NP is 200 and max generation GEN = 2500.Readers are referred to [26] for details of the designed DEalgorithm and its comparative results on benchmark testproblems. The working procedure of the designed COMDEalgorithm is presented inAlgorithm 2.Theparameter settingsrequired for COMDE are shown in Table 3.
7.3. Handling of Integer Variables. In its canonical form, thedifferential evolution algorithm and COMDE algorithm are
Computational Intelligence and Neuroscience 11
(1) Begin(2) 𝐺 = 0, GEN = 2500, NP = 200.(3) Create a random initial population ��
𝐺
𝑖∀𝑖, 𝑖 = 1, . . . ,NP
(4) Evaluate 𝑓(��𝐺𝑖), 𝑐V(��𝐺
𝑖), ∀𝑖, 𝑖 = 1, . . . ,NP
(5) Determine 𝑥𝐺best and 𝑥𝐺
worst based on 𝑓(��𝐺
𝑖) and 𝑐V(��𝐺
𝑖), ∀𝑖, 𝑖 = 1, . . . ,NP
(6) For 𝐺 = 1 to GEN Do(7) CR = 0.95 + (0.5–0.95) ⋅ (1 − 𝐺/GEN)4
(8) Factor ={{{
{{{
{
𝐹final + (𝐹initial − 𝐹final) ∗ (1 −
𝐺
GEN)
𝑘
, 0 < (
𝐺
GEN) ≤ 𝑅
𝐹final, (
𝐺
GEN) > 𝑅
(9) For 𝑖 = 1 to NP Do(10) If (rand[0, 1] ≤ 0.5) Then (Use New Directed Mutation Scheme)(11) Select randomly 𝑟1 = best = worst = 𝑖 ∈ [1,NP](12) 𝐹
(30) End If(31) End For(32) End If(33) Verify boundary constraints(34) If (��𝐺+1
𝑖is better than ��
𝐺
𝑖(based on the three selection criteria))Then
(35) ��𝐺+1
𝑖= ��𝐺+1
𝑖
(36) Else(37) ��
𝐺+1
𝑖= ��𝐺
𝑖
(38) End If(39) Determine 𝑥𝐺+1best and 𝑥
𝐺+1
worst based on 𝑓(��𝐺+1
𝑖) and 𝑐V(��𝐺+1
𝑖), ∀𝑖, 𝑖 = 1, . . . ,NP
(40) End For(41) 𝐺 = 𝐺 + 1
(42) End For(43) End
Algorithm 2: Description of COMDE algorithm.
only capable of optimizing unconstrained problems withcontinuous variables. However, there are very few attemptsto transform the canonical DE and proposed COMDE algo-rithms to handle integer variables [34–37]. In this research,only a couple of simple modifications are required: the newgeneration of initial population and boundary constraintsverification, the proposed novel mutation operation, and thebasic mutation schemes use rounding operator, where theoperator round(𝑥) rounds the elements of 𝑥 to the nearest
integers. Therefore, the initialization and mutations are asfollows:
(i) Initialization and boundary constraint verification:𝑥0
𝑖𝑗= round(𝑙𝑗 + rand𝑗 ∗ (𝑢𝑗 − 𝑙𝑗)).
(ii) New directed mutation: 𝑢𝐺+1𝑖𝑗
= round(𝑥𝐺𝑟1
+ 𝐹𝑙 ∗
(𝑥𝐺
best − 𝑥𝐺
worst)).(iii) Basic mutation: 𝑢𝐺+1
𝑖𝑗= round(𝑥𝐺
𝑟1+ 𝐹𝑔 ∗ (𝑥
𝐺
𝑟2− 𝑥𝐺
𝑟3)).
12 Computational Intelligence and Neuroscience
Table 3: Parameter settings.
Control parameter Actual valuesPopulation size (NP) 200Maximumgenerations (GEN) 2500
Local scaling factor(𝐹𝑙) Uniform random number [0.4, 0.6]
Global scaling factor(𝐹𝑔)
Uniform random number (−1, 0) ∪ (0, 1)
Table 4: Optimal values of nonzero design variables for the casestudy.
Number Decision variable Optimal solution1 𝑥
1,𝑈
𝐸,𝑏,𝑇136,125
2 𝑥1,𝑈
𝐸,𝑔,𝑇72,903
3 𝑥1,𝑈
𝐸,𝑏,𝑀7,425
4 𝑥1,𝑈
𝐸,𝑏,𝑆74,250
5 𝑥1,𝑈
𝐸,𝑔,𝑀3,977
6 𝑥1,𝑈
𝐸,𝑔,𝑆39,765
7 𝑥1,𝑈
𝐸,𝑏,𝐴54,450
8 𝑥1,𝑈
𝐸,𝑔,𝐴29,161
9 𝑥1,𝑈
𝐺,𝑏,𝑀7,241
10 𝑥1,𝑈
𝐺,𝑏,𝑆79,736
11 𝑥1,𝑈
𝐺,𝑏,𝐴29,156
12 𝑥1,𝑈
𝐺,𝑔,𝑀6,047
13 𝑥1,𝑈
𝐺,𝑔,𝑆78,937
14 𝑥1,𝑈
𝐺,𝑔,𝐴33,145
𝑧 = sum of nonzero deviation variables 503,555
8. Problem Solution
The proposed GPM for the admission problem discussedin the previous sections has been tested. The experimentswere carried out on an Intel Pentium core 2 due processor2200MHz and 2GB RAM. COMDE algorithm is coded andrealized inMATLAB.The best result in terms of the objectivefunction value and the optimal decision variables is given inTable 3. 30 independent runs are performed and statisticalresults are provided including the best, median, mean, andworst results and the standard deviation is presented inTable 4. The convergence graph corresponding to the bestobjective function value𝑓(𝑥) of the best run of the case studyagainst Total Number of Function Evolutions (TNFE) of theCOMDE is shown in Figure 3.
In this goal programming model, all the goals for theenrollment part are consideredwith the same importance andare given equal weights of 1 in the objective function. Theincrease in the graduation rate is given higher priority thanthe number of the available jobs.
In the optimal solution, some of the goals are satisfiedwhile some others are over- or underachieved:
Table 5: The statistical results of COMDE on the test problem.
Best Median Mean Worst Std.Test problem 503,555 503,567 503,573 503,595 13.0026
(i) Goals numbers 3, 4, 5, 6, 7, and 8 for enrollment partand goals (19–24) for the graduation part are exactlysatisfied.
(ii) Goals numbers 1, 2, 9, and 16 for the enrollment partare underachieved.
(iii) Goals numbers 10, 11, 12, 13, 14, and 15 for theenrollment part and goals (25–30) for the graduationpart are overachieved.
So, we have the following:
(i) Exact satisfaction of the control for the educationtracks, total number of enrollment students in rela-tion to high school graduates, and number of pop-ulation and student-to-faculty members for medicalboys section.
(ii) Underachievement of the enrollment rate increase,student-to-faculty members for medical girls section,and budget.
(iii) Overachievement of student-to-faculty members forall specialties, boys and girls, except for student-to-faculty members for medical specialties, boys andgirls sections.
(iv) Exact satisfaction of the increase in the numberof graduated students and overachievement of theavailable number of jobs.
From Table 4, it can be seen that the best results obtainedby COMDE are the optimal feasible solution as the con-straints are satisfied. Besides, from Table 5, COMDE isunable to reach the best solution consistently in all runsas the problem is very difficult as discussed previously.However, the median, mean, and worst solutions obtainedby COMDE are not far from the best with small stan-dard deviations which prove COMDE is robust technique.Moreover, convergence behavior is another important factorto be considered in solving optimization problems usingevolutionary algorithms. From Figure 3, it can be deducedthat the optimal solution can be reached using around 70%of total number of function evaluations which shows thatCOMDE is an efficient algorithm with rapid convergencespeed. Based on the above results and analysis, it can beconcluded that COMDE algorithm has a remarkable abilityto solve considered nonlinear integer GP problems with aperfect performance in terms of high quality solution, rapidconvergence speed, efficiency, and robustness.
9. Conclusions and Points forFuture Researches
The national strategic plan for the admission capacity prob-lem in higher education can be satisfactorily designed using
Computational Intelligence and Neuroscience 13
5.01
5.015
5.02
5.025
5.03
5.035
5.04
Best
func
tion
valu
e
0.5 1 1.5 2 2.5 3 3.5 4 4.5 50Number of function evaluations
×105
×105
Figure 3: Convergence graph (best curve) of COMDE on the testproblem.
a general nonlinear integer goal programming model. Themodel is formulated in a general form to satisfy the mainobjectives stated in the development plan of a country or aninstitution.Themain objectives for themodel are to copewiththe increasing demand for higher education in the countryand to satisfy the job market requirements, fair studentsatisfaction, and control over the education tracks (medical,science and engineering, and arts) under the limitation ofavailable resources. The procedure of solution is a stepwiseone; the mode is initially formulated for the first year of theplan. The obtained results are entered as input for the secondyear, and so on, till reaching the last year of the plan.
A recent novel differential evolution algorithm is used tofind the optimum solution for many scenarios representingdifferent priorities for the problem goals.
As future researches, it is proposed to consider thefollowing points:
(i) To apply the samemodel for different countries takinginto consideration the special goals stated in theirstrategic development plans.
(ii) To formulate a multiobjective mathematical pro-gramming model for the same problem and considerdifferent utility functions and paretooptimal solu-tions.
(iii) To design a complete decision support system withuser-friendly interfaces to facilitate the task for thedecision-makers.
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper.
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