A near Pareto optimal approach to student-supervisor allocation with two sided preferences and workload balance
Victor Sanchez-Anguix, Rithin Chalumuri, Reyhan Aydogan, Vicente Julian
AA near Pareto optimal approach to student-supervisorallocation with two sided preferences and workloadbalance
Victor Sanchez-Anguix a,c,b , Rithin Chalumuri c , Reyhan Aydo˘gan d,e , VicenteJulian f a Florida Universitaria, Carrer del Rei en Jaume I, 2, 46470, Catarroja, Valencia, Spain b Universidad Isabel I, Calle de Fern´an Gonz´alez 76, 09003, Burgos, Spain c Coventry University, School of Computing, Electronics and Mathematics, Gulson Rd, CV12JH, Coventry, United Kingdom d ¨Ozye˘gin University, Department of Computer Science, Istanbul, Turkey e Delft University of Technology, Interactive Intelligence Group, Delft, The Netherlands f Universitat Polit`ecnica de Val`encia, Departamento de Sistemas Inform´aticos yComputaci´on, Cam´ı de Vera s/n, 46022, Valencia, Spain
Abstract
The problem of allocating students to supervisors for the development of apersonal project or a dissertation is a crucial activity in the higher educationenvironment, as it enables students to get feedback on their work from an expertand improve their personal, academic, and professional abilities. In this article,we propose a multi-objective and near Pareto optimal genetic algorithm for theallocation of students to supervisors. The allocation takes into considerationthe students and supervisors’ preferences on research/project topics, the lowerand upper supervision quotas of supervisors, as well as the workload balanceamongst supervisors. We introduce novel mutation and crossover operators forthe student-supervisor allocation problem. The experiments carried out showthat the components of the genetic algorithm are more apt for the problemthan classic components, and that the genetic algorithm is capable of producingallocations that are near Pareto optimal in a reasonable time.
Keywords: genetic algorithms, student-project allocation, matching, Paretooptimal, artificial intelligence
Email addresses: [email protected] (Victor Sanchez-Anguix), [email protected] (Victor Sanchez-Anguix), [email protected] (VictorSanchez-Anguix), [email protected] (Rithin Chalumuri), [email protected] (Reyhan Aydo˘gan), [email protected] (VicenteJulian)
Preprint submitted to Elsevier December 18, 2018 a r X i v : . [ c s . A I] D ec . Introduction Every year in higher education (HE) institutions, students undertake indi-vidual projects that are supervised by a tutor that offers academic advice andguidance, either as an undergraduate or master dissertation, as part of theircoursework, or simply as a summer research project. Students are usually allo-cated to supervisors for their projects by means of a centralized human decisionmaker or by means of interactions between students and staff members. Thedecision makers have to take into consideration the preferences of both studentsand supervisors with respect to the conduct of the project, as well as depart-mental constraints such as minimum and maximum levels of workload (in termsof supervision) for each supervisor. This situation results in an extremely timeconsuming process, and a suboptimal allocation due to a large and complexsearch space faced by human decision makers. Automating this process byapplying artificial intelligence techniques may enhance the process in terms ofsatisfaction and performance of students with these individual projects.In this article, we present a genetic algorithm for matching students to su-pervisors according to both students’ and supervisors’ preferences and the con-straints of the department. The rationale behind this problem is matchingan appropriate student with a supervisor for the development of an individ-ual project. The problem of matching students to supervisors, or students toprojects [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13], is a subclass of the widerproblem of matching between two sets, one of the most studied fields in com-puter science due to its applications to a wide range of domains such as thehospital/residents (HR) or the college admission (CA) problem [14, 15, 16].Particularly, the student-supervisor allocation problem solved in this article canbe considered as an instance of the CA problem with lower and upper quotas,where the colleges are the supervisors, both colleges and students (i.e., supervi-sors and students in our case) have some representation of preferences on eachother for the conduct of a project, and the minimum and maximum quotas arethe minimum and maximum number of students to be supervised by staff mem-bers. In this situation, it has been shown that there is no guarantee for a stableallocation to exist and even looking for a near-stable allocation is a NP-hardcomplex problem [15, 16].In order to tackle the complexity mentioned above, in this article we pro-pose a multi-objective and Pareto optimal genetic algorithm (GA). The mainhighlights of the GA proposed in this article are: (i) it takes into considerationboth the preferences of the students and the supervisors with respect to typeof project to undertake/supervise; (ii) it considers the constraints on the indi-vidual minimum and maximum supervision workload of each supervisor; (iii) itaims to provide a fair and balanced allocation in terms of the workload for eachsupervisor; (iv) it provides multiple near-optimal solutions considering both Stability defined as the lack of incentive for any pair of student and college to changetheir current allocation in favor of one that allocates them together.
2. Related work
The problem of allocating the students to supervisor is a one-to-many match-ing where we allocate only one supervisor to each student while more than onestudent can be assigned to a supervisor. As mentioned above, this particular-ity makes it similar to the college admission (CA) and the hospital/residents(HR) problem, two well-known one-to-many matching problems from the pointof view of theoretical computer science. In the HR problem, each resident has aranked list of preferences on the hospitals they may be assigned to, and hospitalsalso have ranked preferences on the residents they may accept. Similarly, in theCA problem, each student has ranked preferences on the colleges they may beaccepted, and each college has ranked preferences on the students that they mayaccept. Both colleges and hospitals may accept more than one student/resident,making it a one-to-many matching.Our student-supervisor matching problem involves a one-to-many matchingwhere supervisors have both lower and upper supervision quotas, and bothsides have preferences on each other. Biro et al. [14] studied the problem of theCA problem with both lower and upper quotas from a theoretical perspective.Differently to our setting, the work presented in [14] allows for colleges to beclosed in case that their minimum acceptance quota is not reached. The authorsfound that, in the presence of both lower and upper quotas, there may not exist astable matching. A matching is considered stable if for every pair of student andcollege not included in the matching, either the student is matched to a collegethat he/she prefers, or the college quota is full with applicants that the universityprefers. In addition to this, determining if a stable matching exists in thissetting is a NP problem. Biro et al. showed that a student oriented polynomialalgorithm and a college oriented polynomial algorithm can be provided in thecase that colleges are organized in nested sets and have common quotas. In our3ork, the minimum supervision quota for each supervisor must be achieved, andall specified staff members participate in the allocation.Later on, Hamada et al. [15, 16] further studied the problem of matchingwith lower and upper quotas, focusing on scenarios where all colleges/hospitalsshould reach their minimum quota in the allocation. In this particular scenario,it is proved that the problem of providing a matching that is as close as possibleto be stable is still a NP-hard problem, but a polynomial time algorithm approx-imation algorithm exists with an approximation guarantee equal to the sum ofthe number of hospitals and residents. This approximation may not be appro-priate for large numbers of hospitals and residents or student and supervisors,as the problem faced in our proposal.The previous findings provide the reader with some background on the com-plexity of the matching problem presented in this article. This complexity,and the need to appropriately tackle large problems motivated the choice ofa metaheuristic instead of a global optimization technique. In the next fewlines, we discuss how our present work compares to other student-project orstudent-supervisor allocation schemes proposed in the literature.Anwar et al. [1] were one of the pioneering authors in providing a computa-tional solution to the student-project allocation problem. The article introducestwo different integer programming models: one to allocate students to projectswhile minimizing the projects supervised by staff members, and another to max-imize the students’ satisfaction according to their preferences on group projectsto be allocated and to be undertaken. In this setting, staff members propose alist of projects and students provide a rank of four projects to be allocated on.Both integer programming models were tested on a real dataset consisting of 60projects, 22 staff members, and 39 students. Similarly, [2] introduces the use ofgenetic algorithms for solving the student-project allocation problem. In theirsetting, students provide a ranked list with their most preferred projects, andeach student is allocated a project from the provided list, with projects beingcarried out individually. The algorithm was tested with real data consisting of25 students and 34 projects, and also with problems created from data providedfrom the OR-library [17]. These models only take into consideration the stu-dents preferences, but they do not consider the staff preferences with regards toprojects and students and the workload of supervisors. In addition to this, theycan only optimize a single objective function which precludes decision makersfrom trading-off between the students’ and the staff preferences.Abraham et al. [3] focus on solving the student-project allocation problemfrom an optimal perspective. The authors assume that a list of projects isprovided by staff members. The students provide a ranked list of their mostpreferred projects, while staff members explicitly rank students that desire tobe allocated to the staff member’s projects. Under this assumption, the authorsprovide two linear algorithms to find stable matching: one from the perspectiveof the students’ preferences, and another from the perspective of the lecturers’preferences. While an optimal solution can be guaranteed employing these algo-rithms, they either provide the optimal solution for the students or the optimalsolution for the staff members, but no trade-off opportunity is provided to deci-4ion makers. In addition to this, the algorithms do not take into considerationthe workload of supervisors, with the possibility of producing unbalanced so-lutions. Finally, it should also be considered that supervisors explicitly rankstudents which may not be feasible if supervisors do not know students, or itmay be unfair for students with lower marks as many will end up in the lastrank positions in lecturers’ preferences.Later on, Manlove & O’Malley [4] study the student-project allocation prob-lem in a scenario where students and supervisors have preferences over a set ofprojects. Both projects and supervisors have capacity constraints. Under theseconditions, the authors prove that stable matchings can have different cardi-nalities, and thus the objective is that of finding the stable matching with amaximum cardinality. Solving this problem is NP hard, but the authors providea student oriented approximation algorithm with a performance guarantee of 2(i.e., only guaranteeing half of the cardinality of the maximum stable matching)and polynomial complexity. Iwama et al. [8] further narrowed down this boundto a range between 1.5 and 1.10. The proposed algorithms focus on optimizingthe students’ preferences, with no explicit consideration of the staff members’preferences, the workload of supervisors, or lower quota constraints.Another genetic approach to the student-project allocation problem was pro-vided by Srinivasan & Rachmawati [5]. The described scenario consists of stu-dents providing a ranked list of projects from a list published every year bylecturers. The problem is tackled as multiobjective optimization problem whereboth the preferences of the students and the departments are taken into consid-eration. In order to compute the preferences of the department, the academicperformance of the students and the workload of supervisors/departments aretaken into consideration. As mentioned, assigning projects on merit may lead toundesirable situations whereby low performing students end up in less attractiveprojects. In addition to this, the model does not support lower and maximumsupervision quotas for lecturers. Finally, it should be highlighted that despitethe fact that multiple objectives are considered, these are aggregated into a sin-gle and final objective function. This requires to compute the GA every timethat the human decision maker desires to trade-off between different objectives.The work presented on [6] proposes the use of goal programming to tackle thestudent-project allocation as a hierarchical multiobjective problem. The maxi-mum priority of the model is maximizing the number of allocated students, andthen it attempts to maximize the students’ preferences and then the academicperformance of allocated students. Again, the model employs academic per-formance to prioritize the departments’ choice, which may be discriminatory.Moreover, the model does not allow to execute trade-offs between the differentobjectives, and it does not guarantee any degree of optimality for each of them.The authors in [7] present an artificial immune system optimization algo-rithm for the student-project allocation problem. More specifically, the authorsmodel a problem where a set of students and projects exist, and students havepreferences on the projects to undertake. In their framework, students must bematched a project, and a project can be matched at most once. The authorsstudy the performance of several mutation operators on the problem, although5hey focus on swapping projects between students based on different criterialike time. As it will be appreciated, our proposed mutation operator takes intoconsideration both swapping (students between supervisors) and transferringoperations (giving a student to another supervisor) and they consider the min-imum and maximum supervision quota of each supervisor.In [9], the authors focus on solving the student-project allocation problemwhere only students’ preferences are present, but supervisors have both lowerand upper supervision quotas. In the article, the authors provide efficient al-gorithms that aim to provide optimal solutions in the context of a single sideoptimization (i.e., students’ preferences). For that, their proposed algorithmsguarantee finding greedy maximum matchings or generous maximum match-ings. The first aims to find the largest matching in terms of the number ofstudents allocated, and maximizing the number of students allocated their firstand most preferred choices. The second aims the largest matching that mini-mizes the number of students allocated their least preferred choices. The workpresented by the authors does not support lecturers’ preferences, and thus opti-mizes a single objective criteria, and matchings found do not necessarily guar-antee matching all of the students to projects/supervisors, something that wehave considered fundamental in our present work.Salami and Mamman propose another genetic algorithm for scenarios wherestudents have complete preferences on supervisors, and supervisors have a max-imum supervision quota [11]. However, there is no consideration on supervisors’preferences or the workload balance for supervisors.In [13] the authors present mixed integer programming models for solving thestudent-project allocation problem with one-sided preferences (i.e., students).Differently to other approaches, students apply for projects in teams and themaximum capacity of projects is defined in number of teams rather than thenumber of individuals. The main focus of the article consists of analyzing dif-ferent fairness metrics from the point of view of the students’ allocation.Recently, Cooper and Manlove [12] have revisited the problem of allocatingstudents to projects, where both students and lecturers have preferences overeach other, and lecturers and projects have upper capacity constraints. Theauthors have provided a 3/2-approximation algorithm capable of calculatingmaximum stable matchings in linear time. It should be considered that thiswork does not include lower quotas and neither introduces fair balancing for thesupervisors’ workload.Our approach is based on students’ and supervisors’ preferences on projecttopics rather than projects. This is an advantage as it does not require lectur-ers to propose projects prior to the allocation and they can be negotiated withstudents according to their research interests. Furthermore, it does not discrim-inate students according to their performance as staff preferences’ are based ontopics rather than students. Most analyzed works only take into considerationthe students’ preferences [1, 2, 7, 9, 11, 13], or they base the department prefer-ences on pure academic merit/opinion [3, 5, 6]. We consider both the students’and the departments’ preferences by adopting a multiobjective approach thatprovides decision makers with flexibility to trade-off between objectives as it6stimates Pareto optimal solutions. The works analyzed in this section are ei-ther single objective (sided) [1, 2, 4, 7, 9, 11, 13] or they adopt a multiobjectivestance by aggregating or prioritizing objectives [5, 6] or by focusing on findingefficient matchings with no lower quotas [4, 8, 12]. In addition to this, we aimto provide a balanced allocation that takes into consideration the workload oflecturers, a characteristic that is only present in [5]. Therefore, the proposedmodel should be more apt for the student-project allocation problem describedin this article.
3. Problem definition
In this section we describe the problem of allocating students to supervisorsfrom a formal perspective. Let S = { s , . . . , s n } and R = { r , . . . , r m } representa set of students and a set of supervisors where n and m denote the number ofstudents and number of supervisors respectively. A matching M is an assignment of students to supervisors, where eachstudent is assigned exactly one supervisor. Without loss of generality, we saythat M ( s i ) represents the supervisor assigned to student s i in M , and M ( r j )represents the set of students assigned to supervisor r j in M . Each supervisor r j has an upper bound supervision quota c j,max , which is normally established bythe head of the department or school. Similarly, each supervisor r j has a lowersupervision quota c j,min , set by the department or school, that determines theminimum number of students that he/she should supervise. This is the case inmany higher education institutions, where supervisors have different teachingloads and therefore, they may be more or less available to supervise students’projects.Given a matching, we say that a supervisor r j is under-subscribed iff c j,min ≤| M ( r j ) | < c j,max , he/she is full iff | M ( r j ) | = c j,max , and he/she is over-subscribed iff | M ( r j ) | > c j,max . We say that a matching M is feasible iff ∀ r j ∈ R , c j,min ≤ | M ( r j ) | ≤ c j,max , i.e. for every supervisor he/she is fullor under-subscribed. As mentioned in the article, we aim to consider the balance of the workloadfor supervisors when constructing a proper allocation with our GA. Therefore,we must provide a formal definition for what we consider workload and howto measure the balance of the workload in a matching M . The workload levelof a supervisor r j in M as l j = | M ( r j ) | c j,max . Namely, that is the ratio of studentssupervised in the matching M over the maximum number of students that can Please note that the definition of matching employed in this paper aligns with the defini-tion employed in the student-project allocation problem. It should be highlighted that this isdifferent to the classic definition of matching in a graph.
7e supervised by r j . Analogously, we can define and L M = { l , . . . , l m } as avector that contains the workload levels for all supervisors in the matching M ,and we define σ L M as the standard deviation of the workload levels of supervisorsin the vector L M . The first step towards evaluating the quality of a matching M is that ofevaluating the individual allocation of student s i to supervisor r j . We define V i,j , as the value given by a student s i to being allocated a supervisor r j , and V (cid:48) j,i as the value given by a supervisor r j to being allocated a student s i .In this work we assume that students cannot explicitly provide a completelist of preferences for supervisors. Even if they could provide a partial list ofsupervisors in rank of preference, the list would be biased to the supervisors thatthey like or the ones that they have met. It is not possible for the students toknow all of the staff members in a relatively large school or department. Thereare different reasons for this. Additionally, students may hesitate to specify theirpreferences on their supervisors/teachers directly due to academic reasons andprivacy issues. However, it is easy for them to specify which topics they wouldlike to work on more. Therefore, we consider that students are able to provide aranked list of k topics to represent their preferences in the problem. Again, weassume that only k explicit preferences can be given as the number of potentialtopics may be extremely large in some areas and that may result in a toocostly elicitation process. Similarly, we can state the same about supervisors.They cannot explicitly rank all of the students as they may not know them.In addition to this, by making supervisors express their preferences in termsof project topics rather than students, we avoid discrimination according toacademic performance.In order to evaluate both the students’ and supervisors’ preferences on topics,we assume that the topics are represented in a tree-like and hierarchical structurewith a common root. In this structure, topics may be further divided intosubtopics and so on, but it is always possible to relate how similar two topicsare by analyzing the tree structure [18]. An example of this tree-like structurecan be observed in Figure 1. In the example, let us assume that a student hasstated that his preferred topic is kw
5. Given the tree structure, one can easilycompare how similar or related other keywords are based on the the numberof common nodes in the tree structure. This is the case, despite the fact thatthe student may have not explicitly provided preferences for other keywords.We assume that each supervisor/student preferences are represented by a listof k different topics from a tree-like and hierarchical structure. A student orsupervisor i describes his preferences with a ranked list KW i = { kw , . . . , kw k } of k topics where ∀ j < k, kw j (cid:31) kw k .We consider that the similarity between a student’s and a supervisor’s pref-erences depends on two factors: the similarity of the keywords provided by bothin their lists, and the position of those keywords in their ranked lists. First, wedefine the similarity between two keywords, and then we define the similaritybetween the positions occupied by two keywords in two ranked lists.8 w2 kw3 kw5 kw6 kw8 kw9kw1 kw4 kw7kw0 Figure 1: Topics organized in a tree-like structure • Keyword similarity:
Let us consider that there is a tree defined by T = ( KW , E ) where KW = { kw , . . . , kw l } is a group of l different nodesthat represent topics in an area of knowledge, and E is a group of edges inthe form ( kw i , kw j ) indicating that kw j specializes the topic in kw i . Thesimilarity of kw j to kw i topics in T is defined as: S T ( kw i , kw j ) = | path ( kw i , T ) ∩ path ( kw j , T ) || path ( kw i , T ) | (1)where path : KW × T → KW is a function that retrieves the path definedfrom the root of the tree T to the node kw (included). As a consequence,we define the similarity of kw j to kw i as the number of common nodes inthe path defined from the root to both topics. Please, the reader shouldbear in mind that this similarity metric is not symmetric, to consider thefact that more specific topics are only fully matched by topics of greateror the same specificity. Lastly, if we assume two lists of preferences KW i and KW j , and a topic kw i ∈ KW i , we define its best matching topic kw ∗ i ∈ KW j as arg max kw j ∈KW j S T ( kw i , kw j ). • Rank similarity:
We define S rnk as the rank similarity between twokeywords in two ranked lists of preferences KW i and KW j . This similaritymetric represents the fact that the order of the topics in both the studentand supervisor’s preferences should matter as it denotes the degree ofinterest of expertise in the topic. For instance, let us assume that thetopic artificial intelligence is defined as the most preferred topic for astudent, and the best matching keyword in a supervisor’s list is machinelearning . However, this topic appears as the last in the list of preferencesfor the supervisor. The student should prefer matching supervisors thathave a closely related topic higher in their rank of preferences as it is themost preferred topic for the student. This fact is reflected by the definition9f the rank similarity: S rnk ( kw i , kw j , KW i , KW j ) = 11 + | pos ( kw i , KW i ) − pos ( kw j , KW j ) | (2)where the function pos returns the position of a keyword in a ranked list ofpreferences, with lower positions representing choices higher in rank. Thissimilarity metric reflects the fact that the positions of the keyword andbest matching keyword in both supervisors and students is important.From this point on, we define the evaluation given by a student s i to asupervisor r j : V i,j . The evaluation given by s i to r j is defined as: V i,j = (cid:88) kw i,r ∈KW i w r × S rnk ( kw i,r , kw ∗ i,r , KW i , KW j ) × S T ( kw i,r , kw ∗ i,r ) (3)where, as mentioned, we define kw ∗ i,r ∈ KW j as the best matching topic to kw i , and w r is a weight indicating the importance of matching the r -th mostimportant preference for the student. This way, we take into considerationthat students may prefer to be matched according to their most preferred topicrather than topics further down in their ranked list of preferences. It should behighlighted that the evaluation given by a supervisor r j to the student s i , V (cid:48) j,i ,can be defined in an analogous way. As we have mentioned in the previous section, in this article we considerboth the preferences of the students and supervisors. This means that thereare two objectives to be maximized and this is a multi-objective optimizationproblem. We describe this optimization problem first from the point of view ofstudents and then from the point of view of the supervisors.
From the point of view of the students, the optimization problem is to max-imize the overall satisfaction of the students from their assigned supervisors.Next, we define the associated optimization problem:max 1 |S| (cid:88) s i ∈S (cid:88) r j ∈R x i,j × V i,j subject to ∀ r j ∈ R , c j,min ≤ (cid:88) s i ∈S x i,j ≤ c j,max (4) ∀ s i ∈ S , (cid:88) r j ∈R x i,j = 10 ≤ x i,j ≤ x i,j is a binary variable that indicates if the student s i has been allo-cated to supervisor r j in matching M . The optimization function, which we aimto maximize, is defined as the mean of the valuation given by students for theirassigned supervisors in matching M . The first constraint forces the optimizationproblem to find a solution where no supervisor r j is over his/her upper boundsupervision quota c j,max , and that a minimum of c j,min students are allocatedto r j . This latter value represents situations where the department establishes aminimum supervision workload for supervisors. The next constraint forces theoptimization problem to assign a student s i to just one supervisor. Finally, thelast constraint defines the domain for the binary variables. As mentioned above, the other optimization problem is defined by the in-terests of the supervisors. In this article, we assume that the interests of thedepartment are (i) to make supervisors more comfortable with their work byassigning a student who is willing to work in areas related to the supervisor’sexpertise; (ii) and to avoid unbalanced solutions where well-known supervisorshave a much higher supervision load than other staff members, as this couldcause friction and envy amongst coworkers. Given these assumptions, we definethe optimization problem faced by the department as follows:max 1(1 + σ L M ) α × |R| (cid:88) r j ∈R | M ( r j ) | (cid:88) s i ∈S x i,j × V (cid:48) j,i subject to ∀ r j ∈ R , c j,min ≤ (cid:88) s i ∈S x i,j ≤ c j,max (5) ∀ s i ∈ S , (cid:88) r j ∈R x i,j = 10 ≤ x i,j ≤ σ L M in M . The greaterthe standard deviation, the greater becomes the penalization factor and the moreis reduced the value of the objective function. As a result, given two allocationswith the same value stemming from the value given by students assigned tosupervisors, the optimization problem prefers solutions with a more balancedworkload level. This avoids situations like the one mentioned above, wherepopular supervisors are highly subscribed while others have a very low workload11evel. The effect of this parameter can be further expanded by the coefficient α , which should penalize allocations with a higher workload unbalance when α >
4. A Pareto optimal genetic algorithm for the student-supervisor al-location problem
Due to the ability of dealing with large search spaces and providing goodsolutions in a reasonable amount of time, we decide to use genetic algorithmsto solve the student-supervisor allocation problem. In this section, we describethe design and implementation of the proposed genetic algorithm. In additionto this, metaheuristics tend to provide a good solution in a reasonable amountof time. Exact methods for non-linear, or even linear problems such as theone presented in Section 3 are known to be costly in time for large searchspaces. Thus, we select genetic algorithms as the method for solving the problempresented in this work.As the reader may have observed, there are two different objective functionsfor the problem. Therefore, in this article we opt for a Pareto optimal geneticalgorithm. Pareto optimal methods allow to retrieve a variety of non-dominatedsolutions which can later be analyzed by a decision maker to trade-off betweenthe different objective functions. In this case, the staff entitled with the task ofallocating students to supervisors can select from a wide range of allocations tobetter reflect the priorities of the students and the supervisors. More specifically,due to the fact that the problem is composed by just two objective functions,we employ a schema inspired by NSGA-II [19], a well-known GA schema forestimating Pareto optimal solutions in multi-objective problems.Next, we define the specific details of the proposed genetic algorithm. First,we explain how chromosome (solutions) are represented in our GA. Next, wedefine the main operators of the proposed GA: crossover operators, and mutationoperator. Finally, we describe the selection mechanism employed and the outlineof the GA.
For this GA, we employ a graph to represent a solution. Formally, a matching M can be represented by means of a bipartite graph G M = ( S , R , E ) where E = { ( s i , r j ) | M ( s i ) = r j } is the set of edges that determine the assignmentof students to supervisors (i.e., an edge is present if a student is matched to asupervisor in M).Figure 2 show an example of how an allocation of 5 students to 3 supervisorsis represented by a bipartite graph. More specifically, in the example provided,the supervisor r supervises student s , supervisor r supervises students s and s , and supervisor r supervises students s and s .In the bipartite graph G M representing an allocation, we define the structureof the allocation as st G M = ( | N ( r , G M ) | , . . . , | N ( r m , G M ) | ) as the number ofneighbors of each supervisor in the bipartite graph (i.e., the number of students12 Figure 2: An example of a matching of 5 students to 3 supervisors represented by a bipartitegraph that each supervisor supervises in the allocation). For instance, in the examplein Figure 2, the structure of the allocation is (1,2,2). As the reader may haveguessed by now, the structure of the allocation is important as it is related tothe workload level of the supervisors and, therefore, to the objective function ofthe supervisors.
We introduce a mutation operator in the context of the student-supervisorallocation problem that employs two actions: swap and transfer . Our swapaction is inspired by the mutation operator in bin packing problems [20]. Simi-larly, the transfer action is inspired on the similar intuition proposed in [21] forbin packing problems.The mutation operator is applied over a single parent and it generates a sin-gle child. For this problem we have designed a special mutation operator thatapplies a series of operations on a parent allocation: swapping of students be-tween supervisors, and transferring of a student from one supervisor to another.The former operation does not change the structure of the allocation (i.e., theworkload of any supervisor), while the latter does by reducing the workload ofa supervisor by one and increasing the load of another supervisor by one.The extent to which a parent changes by a single mutation operation isdefined by the mutation ratio p mt , that represents the probability of mutatingan edge in a graph G M . The type of operation that is applied over an edge thatis to be mutated is controlled by p sw , which controls the probability of applyinga swapping operation between two supervisors. An outline of the mutationoperator can be found in Algorithm 1. The operator iterates over edges in thebipartite graph and attempts to perform an operation over an edge in the graphwith a probability of p mt . In case that a transfer operation is possible for theedge (i.e., the supervisor has more than the minimum quota established by thedepartment), it selects a random supervisor that can take students (i.e., under-subscribed) and performs the operation with a probability of 1 − p sw (lines 1 to9). Otherwise, the operation selected is a swap between supervisors (lines 10 to135), which is always possible in feasible solution as the structure of the solutiondoes not change. Algorithm 1:
Outline of the mutation operator
Input: G M = ( S , R , E ) : A bipartite graph representing a feasible allocation Output: G M (cid:48) = ( S , R , E (cid:48) ) : A new bipartite graph representing a feasibleallocation G M (cid:48) = G M ; U = under-subscribed( G M (cid:48) ); for ( s i , r j ) ∈ E (cid:48) doif random () ≤ p mt thenif random () > p sw ∧ N ( r j , G M (cid:48) ) > c j,min ∧ U (cid:54) = ∅ then /*Transfer operation*/ ; r q = random choice ( U ); E (cid:48) = ( E (cid:48) ∪ { ( s i , r q ) } ) − { ( s i , r j ) } ; U = under-subscribed( G M (cid:48) ); else /*Swap operation*/ ; r q = random choice ( R ); s p = random choice ( N ( r q , G M (cid:48) )); E (cid:48) = ( E (cid:48) ∪ { ( s i , r q ) , ( s p , r j ) } ) − { ( s i , r j ) , ( s p , r q ) } ; endendend As the structure of the allocation is important for the objective function ofthe supervisors, we have devised two crossover operators for matching problemsthat preserve the allocation structure of the parents as much as possible. Morespecifically, these two crossover operators take two parents as input and producea child as a result. Both operators preserve the original allocation structure ofone of the parents; however, the second approach may end up adding new geneticmaterial not present in any of the parents whereas the first one does not. Asa result, the latter crossover operation may induce in additional exploration asnew assignments of students to supervisors. We call the first genetic operator asthe
Hopcroft-Karp genetic operator as it is based on the popular algorithm tofind maximum cardinality matchings in bipartite graphs [22], while the secondreceives the name of greedy structural preservation crossover due to its greedynature for selecting genetic material from parents. Next, we define both indetail.
As we mentioned, this crossover operator generates a new child from twoparents. In order to do so, the new solution inherits the allocation structure14f one of the parents, and it exclusively employs genetic material from the twoparents to generate the child.The outline of this genetic operator is as follows. An example of the appli-cation of this genetic operator to two parents can be found in Figure 3, whilethe formalization of the operator can be found in Algorithm 2: • Merging parents:
This step can be found in Figure 3 (a) and lines 1-10 of Algorithm 2. First of all, we generate a new graph as a result ofmerging both graphs by keeping the set of students and supervisors. Theselection of this structure is proportional to its impact in the objectivefunction of the supervisors. In addition to this, the structure of one ofthe two parents is inherited as a goal for the new bipartite graph. In theexample in Figure 3, the structure of the first parent is chosen. • Transforming graph:
The description of this step can be found in lines11-14 of Algorithm 2 and an example can be found in Figure 3 (b). Themerged graph is transformed into a new bipartite graph whose set of su-pervisors contains a copy of each original supervisor for each student thathe/she should supervise according to the inherited structure. For instance,in the example in Figure 3 (b), there are two copies of the original super-visor r ( r , and r , ) because two students should be assigned to thesecond supervisor in the new allocation. Similarly, the same happens forsupervisor r ( r , , r , ). • Hopcroft-Karp:
Lines 15-16 of Algorithm 2 and Figure 3 (c) representthis step. The Hopcroft-Karp algorithm [22] is applied on the transformedgraph to find a maximum cardinality matching. As the merged graphcontains at least a perfect matching (i.e., one of the two original parents),then the maximum cardinality matching is a perfect matching. • Transforming back:
Finally, the perfect matching is transformed backto the original supervisor set by merging those nodes that represent thesame supervisor. As a result of this process, a new child is generatedthat inherits the structure of one of the two parents and it introduces nonew genetic material. This can be found in lines 17-19 of Algorithm 2 andFigure 3 (d). Note that the structure of the allocation is the same with oneof the parents; however, it does not mean that the allocation is exactly thesame with the chosen parent’s allocation. As seen from the given example,the resulted allocation is different than the parent allocations (e.g. s3 isassigned to r1 in the chosen parent allocation while it is assigned to r2 inthe child allocation).The theoretical complexity of this crossover operator is determined by thecomplexity of each of its individual steps. In order to merge both parents, anew set of edges must be created which consists of all the edges in both parents.As the number of edges in each parent is exactly |S| then the cost of this stepis O ( |S| ). Transforming the graph requires to create a new set of supervisorsthat has exactly as many supervisors as students ( O ( |S| )) and creating a new15 lgorithm 2: The Hopcroft-Karp crossover operator
Input: G M = ( S , R , E ) : A bipartite graph representing a feasible allocation; G M = ( S , R , E ) : A bipartite graph representing a feasible allocation; Output: G M (cid:48) = ( S , R , E (cid:48) ) : A new bipartite graph representing a feasibleallocation /* Merge graphs */ ; G = ( S , R , E = E ∪ E ); /* Inherit one of the structures */ ; p = σ M ) α ; p = σ M ) α ; if random () ≤ p p + p then st G M (cid:48) = {| N ( r , G M ) | , . . . , | N ( r m , G M ) |} ; else st G M (cid:48) = {| N ( r , G M ) | , . . . , | N ( r m , G M ) |} ; end /* Transform graph */ ; R tr = { r j,l | r j ∈ R ∧ l ≤ st G M (cid:48) ( r j ) } ; E tr = { ( s i , r j,l ) | (( s i , r j ) ∈ E ∪ E ) ∧ r j,l ∈ R tr } ; G tr = ( S , R tr , E tr ) ; /* Apply Hopcroft-Karp algorithm */ ; E hp = hopcroft karp ( G tr ); /* Transform back to original representation */ ; E (cid:48) = { ( s i , r j ) | ∃ ( s i , r j,l ) ∈ E hp } ; G (cid:48) = ( S , R , E (cid:48) ) set of edges that is at most O ( |S| ). The most expensive step is applying theHopcroft-Karp algorithm which has a complexity of O ( |S| (cid:112) |S| ) in the worstcase. However, some recent studies show that in the average case the Hopcroft-Karp algorithm has a complexity of Θ( |S| log |S| ) for random sparse bipartitegraphs [23]. The bipartite graphs generated by the merge operation will resultin graphs where students have at most two neighbors (i.e., the student has adifferent supervisor in both parents). Therefore, we expect that in practice thecost of this step will be closer to the Θ( |S| log |S| ) complexity. The final steprequires iterating over resulting edges in the perfect matching which is exactly O ( |S| ). Therefore, the complexity of this operator is O ( |S| (cid:112) |S| ) in the worstcase and we expect it to be Θ( |S| log |S| ) in the average case. In both cases, thecomplexity is quasi-linear. The greedy approach preserves the structure of one of the two parents, whichis randomly inherited based on the impact of the structure on the fitness of thesupervisors. Differently to the
Hopcroft-Karp crossover operator, this crossoveroperator may introduce new genes that are not present in any of the two parents.Nevertheless, the operator aims to keep original genetic material as much aspossible. As a trade-off, the computational complexity of this operator is lower16
MERGE s1s2s3s4s5 r1r2r3 (a) s1s2s3s4s5 r1r2r3 s1s2s3s4s5 r1r2r3 s1s2s3s4s5 r2,2r3,2s1s2s3s4s5 r1r2,1r3,1
TRANSFORM (b) s1s2s3s4s5 r2,2r3,2s1s2s3s4s5 r1r2,1r3,1
HOPCROFTKARP s1s2s3s4s5 r2,2r3,2s1s2s3s4s5 r1r2,1r3,1 (c) s1s2s3s4s5 r2,2r3,2s1s2s3s4s5 r1r2,1r3,1 s1s2s3s4s5 r1r2r3s1s2s3s4s5 r1r2r3
TRANSFORMBACK (d)
Figure 3: Steps of the
Hopcroft-Karp crossover operator (in separate boxes). The order of thesteps is read left to right. than that of the
Hopcroft-Karp as it takes a greedy approach. The general ideabehind this method is locking edges that will be part of the resulting matching,and removing those that are to be discarded.Next, we describe the outline of this genetic operator in more detail. Figure4 shows how the operator is applied over a particular example, while Algorithm3 depicts the specific details of the operator in more detail. The general stepsof the crossover operator are: • Merging parents:
This step can be found in the top left box of Figure4 (a) and lines 1-11 of Algorithm 3. It is equivalent to the merging stepsin
Hopcroft-Karp crossover operator. In addition to inheriting the struc-ture of one of the two solutions, the method initializes a counter for eachsupervisor that contains the number of edges that have been locked forthe final allocation so far in the process. In Figure 4 (a), the allocationstructure inherited is that from the first parent. • Simplify:
The details of this step can be found in Algorithm 3 from lines12 to 20, and an example applied over a real graph is observable in Figure4 (b). The merged bipartite graph is simplified. The simplification processlocks those edges that have a student with a single possible supervisor. Forinstance, in Figure 4 (b), this corresponds to edges starting from students s and s . The locked edges will be part of the final allocation and countersare updated for supervisors whose edges have been locked ( r and r ). Incase that one of the supervisors reaches the desired workload level, allother unlocked edges involving that supervisor will be removed from themerged graph. This step is repeated until the graph cannot be further17implified by this method. • Locking and removing edges:
This step corresponds to lines 21 to 30in Algorithm 3, and Figure 4 (c). An unlocked edge is randomly chosenfrom the current graph (e.g., edge ( s , r ) in Figure 4 (c)) and it is lockedto be part of the final allocation. As the edge has been locked, the numberof locked edges for the supervisor is increased. Any other edges incidentin the selected student are removed from the merged graph (e.g., edge( s , r ) in Figure 4 (c)). In case that the supervisor has reached the desiredworkload level, unlocked edges incident in the supervisor are removed fromthe graph (e.g., edges ( s , r ) and ( s , r ) in Figure 4 (c)). This step isrepeated while there are no more unlocked edges. • Adding edges:
This last step is represented in lines 31 to 39 of Algorithm3 and Figure 4 (e). Once there are no more unlocked edges, the opera-tor checks if there are any unmatched students and supervisors. If thereare unmatched vertex, then the operator randomly assigns students to su-pervisors while following the desired workload level in the allocation, andconsidering the number of locked edges for each supervisor. The processof adding edges is repeated until there are no more unmatched students.The process described above ends up with a feasible allocation which hasinherited the structure of one of the two parents. The complexity of the op-erator is straightforward. As discussed, the merging process has a complexityproportional to O ( |S| ). In the worst case, the simplify step will be applied asmany times as students in the problem (i.e., merging the same parents) whichgives a worst case cost of O ( |S| ). The lock and remove step will be applied asany times as edges in the merged graph, which will be O ( |S| ) in the worst case.Then, the final step adds one edge per remaining unassigned student. This laststep will never be more costly than O ( |S| ). Therefore, the cost of this operatoris linear with the number of students in the problem. The selection mechanism in this GA is employed to determine the parentsthat will take part in the crossover operation. More specifically, we run randomtournaments [24] between solutions in the population until we have selected anumber of pairs that is equal to half of the current population.As this is a multi-objective optimization problem, the comparison carried outin the tournament is determined by the solution that has a lower nondominatedrank or the one that has a higher crowding distance in case of both solutionshaving the same nondominated rank. The nondominated rank of a solution isdetermined when calculating the different Pareto frontiers in the population,and it is related to the number of solutions that dominate the specific solution.On the other hand, the crowding distance makes sure that the solutions arewell-spread on the Pareto frontier. The details of these metrics can be found in[19]. 18
MERGE s1s2s3s4s5 r1r2r3
1 02 02 0 (a) s1s2s3s4s5 r1r2r3 s1s2s3s4s5 r1r2r3
SIMPLIFY s1s2s3s4s5 r1r2r3
1 02 12 1 s1s2s3s4s5 r1r2r3 (b)
LOCK ANDREMOVE s1s2s3s4s5 r1r2r3
1 02 12 1 s1s2s3s4s5 r1r2r3 s1s2s3s4s5 r1r2r3
1 12 12 1 s1s2s3s4s5 r1r2r3 (c)
LOCK ANDREMOVE s1s2s3s4s5 r1r2r3
1 12 22 1 s1s2s3s4s5 r1r2r3s1s2s3s4s5 r1r2r3
1 12 12 1 s1s2s3s4s5 r1r2r3 (d)
ADD EDGES s1s2s3s4s5 r1r2r3
1 12 22 2 s1s2s3s4s5 r1r2r3s1s2s3s4s5 r1r2r3
1 12 22 1 s1s2s3s4s5 r1r2r3 (e)
Figure 4: Steps of the greedy structural preservation crossover operator (in separate boxes).The order of the steps is read left to right. lgorithm 3: The greedy structural preservation crossover operator
Input: G M = ( S , R , E ) : A bipartite graph representing a feasible allocation; G M = ( S , R , E ) : A bipartite graph representing a feasible allocation; Output: G M (cid:48) = ( S , R , E (cid:48) ) : A new bipartite graph representing a feasibleallocation /* Merge graphs */ ; G (cid:48) = ( S (cid:48) = S , R (cid:48) = R , E (cid:48) = E ∪ E ); /* Inherit one of the structures */ ; p = σ M ) α ; p = σ M ) α ; if random () ≤ p p + p then st G M (cid:48) = {| N ( r , G M ) | , . . . , | N ( r m , G M ) |} ; else st G M (cid:48) = {| N ( r , G M ) | , . . . , | N ( r m , G M ) |} ; end L = ∅ /* Initializing locked edges */ ; /* Simplify graph */ ; foreach { ( s i , r j ) | | N ( s i , G M (cid:48) ) | = 1 } do L = L ∪ { ( s i , r j ) } ; l j = l j + 1 /* Update locked edges counter for supervisor j */ ; /* If supervisor has desired workload level, then remove non-locked edges*/ ; if l j = st G M (cid:48) ( j ) then E (cid:48) = E (cid:48) − { ( s u , r j ) | ( s u , r j ) / ∈ L} endend /* Locking and removing edges */ ; while E (cid:48) − L (cid:54) = ∅ do ( s i , r j ) = random choice ( E (cid:48) − L ); E (cid:48) = E (cid:48) − { ( s i , r l ) | r l (cid:54) = r j } /* Remove other edges incident in the student*/ ; L = L ∪ { ( s i , r j ) } ; l j = l j + 1; if l j = st G M (cid:48) ( j ) then E (cid:48) = E (cid:48) − { ( s u , r j ) | ( s u , r j ) / ∈ L} endend /* Adding edges to complete graph */ ; S re = { s i | | N ( s i , G M (cid:48) ) | = 0 } ; R re = { r j | l j (cid:54) = st G M (cid:48) ( r j ) } ; while S re (cid:54) = ∅ do s i = random choice ( S re ); r j = random choice ( R re ); L = L ∪ { ( s i , r j ) } ; update ( S re , R re ); end .5. Evolution schema As mentioned, the outline of the genetic algorithm is inspired by NSGA-II[19]. The details of the GA can be found in Algorithm 4. The genetic algorithminitializes a population of pop max random feasible solutions (line 1). Then, themain loop of the genetic algorithm runs for a fixed number of iterations (lines4-20), determined by the parameter it max .In the main loop, the genetic algorithm calculates the successive Pareto op-timal frontiers in the current population ( P , line 6): calculating the first Paretooptimal frontier, removing the Pareto frontier from the set and calculating anew one following this process until no more frontiers can be calculated. Thegenetic algorithm limits the number of solutions in the population by fillingthe new population ( P new ) with solutions from the first to the latest Paretooptimal frontiers (lines 8-15). Then, each solution in the resulting population ismutated, and the crossover operator is applied over solutions selected by tour-nament selection (lines 16-19). Lines 21 and 22 calculate the resulting frontiersafter applying genetic operators in the last iteration. Algorithm 4:
The proposed Pareto optimal genetic algorithm P = P new = initialize ( pop max ); it = 0; P off = ∅ ; while it < it max do P = P new ∪ P off ; F = calculate frontiers ( P ); P new = ∅ ; foreach f ∈ F doif |P new | + | f | ≤ pop max then P new = P new ∪ f ; else P new = P new ∪ select ( pop max − |P new | , f );break; endend P mut = mutation ( P ); P (cid:48) = tournament selection ( P ); P cr = crossover ( P (cid:48) ); P off = P mut ∪ P cr ; end P = P new ∪ P off ; F = calculate frontiers ( P );
5. Experiments
In order to validate the performance of the proposed genetic algorithm, wecarry out a series of practical experiments. These experiments aim to study the21mpact of the different elements of the genetic algorithm, as well as the overallperformance of the genetic proposal. First, we provide a brief analysis of thereal data collected from the student-supervisor allocation process at CoventryUniversity, as this data is employed to create real allocation problems that willbe employed to validate the performance of the genetic proposal. Then, weempirically analyze the impact of the mutation operator on the performance ofthe GA by studying the appropriate degree of mutation rate and the importancegiven to exploring the structure of the allocation rather than the allocationitself. After that, we analyze the empirical complexity of the
Hopcroft-Karp and the greedy structural preservation crossover operator, and we compare theiroptimization performance with classic crossover operators. Finally, we comparethe performance of the proposed genetic algorithm with that of global optimaloptimization methods to assess the quality of the solutions found by the GA.
In order to test the genetic algorithm in a realistic setting, we collected realdata from undergraduate students and staff members that participate in theundergraduate dissertation module for computing related degrees at CoventryUniversity. The preferences of students and staff members were elicited byallowing individuals to specify, in order, their k = 5 most preferred topics inthe 2012 ACM Computing Classification System . This taxonomy provides atree-like and hierarchical classification of areas in computing, as needed by ourfitness functions, and it is a well-known system employed to categorize researchpapers in computing.A total of of 195 students’ preferences and 33 supervisors’ preferences werecollected. This dataset contains real preferences of students on computingareas for their undergraduate dissertations, as well as the preferences of staffmembers on research areas where they would like to supervise students on.By analyzing the preferences of both students and supervisors, one can ob-serve that there are some differences. For instance, Figure 5 analyzes the distri-bution of the top 10 topics selected by students and supervisors when focusingon the third level of the path defined by the topics selected by both populations.As one can observe, some topics that are popular amongst students like Softwarecreation and management are not as popular for supervisors, while some pop-ular topics amongst students like
Electronic commerce are not even present inthe top 10 third level topics for supervisors. Therefore, there is conflict betweenthe students’ and supervisors’ preferences with respect to dissertation areas.In Figure 6, we analyze the level or depth of the topics provided by super-visors and students. As we can see, there are divergences with respect to thespecificity of topics. The reader can observe that while supervisors were moregeneric with their provided topics, students were more prone to provide a fine-grained topic for their dissertations. Therefore, we can conclude again that the The dataset is available at http://sanchez-anguix.com/index.php/research/ igure 5: Distribution of the top 10 third level topics in the 2012 ACM Computing Classifi-cation System for the topics selected by students (left) and supervisors (right)Figure 6: Level of the topics selected by students (left) and supervisors (right) from the 2012ACM Computing Classification System optimization problem is complex due to the diversity of preferences.It should be highlighted that the preferences contained in this dataset wereemployed to generate the student and supervisor profiles in the subsequent ex-periments. As a first step to analyze the performance of the proposed genetic algorithm,we studied the impact of the mutation operator and its parameters on theproblem. To be specific, we studied what the impact of the parameters p sw and p mt is on the general performance of the GA. For this matter, we created aexperiment as follows: • We created 5 problems consisting of 150 students and 30 supervisors fromthe collected dataset. The minimum workload of supervisors c j,min wasset to 1 student (i.e., a supervisor will advise at least one student) and theupper bound supervision quota c j,max of each supervisor was generatedfrom a uniform distribution U (4 , • We set α = 2 to highly penalize solutions with a high standard deviationfor the workload level of supervisors. • The weights of topics’ ranks in V i,j were set to (0.561,0.258,0.129,0.064,0.032)respectively, following an exponential decreasing function. This way, we23 igure 7: An example of the hypervolume between a reference point and the estimated Paretooptimal frontier (left) and the real Pareto optimal frontier (right). take into consideration the fact that the disappointment of being matchedon the second topic over the first topic is not linearly related to the dif-ference of being matched on the last topic over the second last topic, as itwas suggested by [2]. • The crossover operation was deactivated to isolate the effect of the muta-tion operation on the performance of the genetic algorithm. • The initial population size was set to 128 solution, and the initial pop-ulation was shared amongst different runs of the same case in order tocompare results on a fair basis. • The maximum number of iterations it max was set to 250 iterations. • The values tested for p mt ranged from 0.05 to 0.5 with increments of 0.05.On the other hand, the values tested for p sw ranged from 0.1 to 0.9 withincrements of 0.1.The metrics employed to study the quality of the Pareto optimal frontierobtained by the different configurations are: • The S metric [25]. This metric takes a reference point above the realPareto optimal frontier provided by the researcher, and it calculates thehypervolume between the estimated Pareto optimal frontier and the ref-erence point. The closest the estimated Pareto optimal frontier is to thereal frontier, the lower the volume will be between the estimated fron-tier and the reference point. Figure 7 shows the hypervolume betweena reference point and the estimated Pareto optimal frontier (left), andthe hypervolume between the reference point and the real frontier. As itcan be observed, the closest the estimated frontier is to the real frontier,the lower the hypervolume will be between the frontier and the referencepoint, with the lowest being when the estimated Pareto optimal frontieris equal to the real one. In the experiments, we take (1.0, 1.0) as thereference point. • The maximum fitness found for the students.24 igure 8: Average maximum fitness for the students (left), for the supervisors (right), andthe average hypervolume (bottom) on the different combinations of the mutation rate ( p mt )and the probability of swapping supervisors ( p sw ) when mutating a gene • The maximum fitness found for the supervisors.In order to decide on the best set of parameters for the mutation operator, wefollowed a grid search strategy on all the possible combinations of p mt and p sw .The results of this experiment can be seen in Figure 8. The left heatmapshows the average maximum fitness for the students, while the right heatmapcontains the average maximum fitness for the supervisors. Finally, the bottomheatmap contains the average hypervolume defined by the reference point andthe estimated Pareto optimal frontiers. All of the three heatmaps show a similartrend. In general, the mutation operator is more effective when a small ratio ofthe genes are mutated (i.e., p mt = 0 . p sw tend to be low and between0.1 and 0.2. Another way to interpret this result is that the mutation opera-tor is more suited to the problem when it explores new allocation structuresinstead of remaining on the same allocation structure. This result is important,as both proposed crossover operators do not explore solutions with a new allo-cation structure and, therefore, the goal of the mutation operator will be thatof introducing new allocation structures into the population. As part of the design of our genetic proposal, we have proposed two newcrossover operators that are specifically designed for the problem of allocatingstudents to supervisors. Next, we study some of the practical properties ofthose operators. More specifically, we will focus on studying the experimental25 igure 9: Average time spent by the the
Hopcroft-Karp and the greedy structural preservation crossover operators (left), and the average ratio of new genetic material (right) introduced bythe greedy structural preservation operator with different number of students supervisors. temporal cost of both crossover operators, as well as identifying the ratio ofnew genetic material introduced by the greedy structural preservation crossoveroperator.In Section 4.3 we studied the worst case temporal complexity of both geneticoperators, with the greedy structural preservation operator having a complexityof O ( |S| ), and the Hopcroft-Karp operator having a worst case complexity of O ( |S| (cid:112) |S| ) and an expected average complexity of Θ( |S| log |S| ). In the followingexperiment we study the experimental time complexity of both operators andcorroborate their adherence to their expected complexities.In this experiment, we ranged the number of students from 50 to 500 withsteps of 50. The number of supervisors was set to one tenth of the number ofstudents. The minimum and maximum supervision quotas of supervisors wereset as described in the previous experiment. We generated one problem for eachnumber of students. For each problem, we generated 1000 pairs of solutionsthat would become parents for the crossover operations. Then, for each numberof students, we measured the average time taken by both crossover operatorsover the available pairs of solutions.The results of this experiment can be found in the left graph in Figure 9.The graph shows the average time spent by the Hopcroft-Karp (blue dots) andthe greedy structural preservation (red dots) operators over allocations with dif-ferent number of students. The dot markers represent the experimental datacollected from the experiment, while the lines show the best fitting functionsfor the experimental points. One can observe that the greedy structural preser-vation operator is generally faster than the
Hopcroft-Karp operator, with thedifferences being greater as the number of students increases. This results isaligned with our initial expectations and the suggested temporal complexityfor both operators, as the
Hopcroft-Karp crossover was expected to behave atmost as an O ( |S| log |S| ) algorithm. The best fitting function by least squaresapproximation for the Hopcroft-Karp operator is a nlogn linearithmic function,and the best fitting function for greedy structural preservation operator is, as26xpected, a linear function. This confirms our initial hypothesis with regardsto the
Hopcroft-Karp operator, with the average time being close to the casewhen the underlying graph is random and sparse bipartite. As a result, theoperator can tackle larger problem sizes with a reasonable time, making it moreapplicable in a realistic context.The next experiment that we carried out over our crossover operators hasthe aim of studying the ratio of new genetic material introduced by the greedystructural preservation operator. As it was mentioned, the operator preservesthe structure of one of the two parents (i.e., the number of students allocated toeach supervisor) but there may be some new genetic material that is not presentin any of the two parents.For this experiment, we ranged the number of students from 50 to 500 withsteps of 50, and the number of supervisors was set to be either one eighth,one tenth, or one twelfth of the number of students. Again, the minimum andmaximum capacities of supervisors were set as described in the other experi-ments. For each combination of number of students and number of supervisorswe generated a random problem. Then, for each problem we generated again1000 pairs of solutions to act as parents for the greedy structural preservation operator. For each combination of number of students and supervisors we mea-sured the average ratio of new genetic material (i.e., number of new genes overthe total number of genes) introduced by the operator over the 1000 crossoveroperations carried out.The results of this experiment can be found in the right graph in Figure9. The first observation that can be made is that, regardless of the proportionbetween the number of students and supervisors, the trend appreciated is sim-ilar and so is the average ratio of new genetic material introduced in the threescenarios. The ratio of new genetic material tends to become smaller as the sizeof the problem becomes larger, with the highest ratios found at small number ofstudents. Despite these ratios being higher with smaller problems, they shouldnot be considered as disruptive with respect to the original parents. In fact,the average ratio of new genetic material in the experiments ranges from ap-proximately 9% to approximately 12% of the genes. Therefore, the new geneticmaterial introduced by the operator only explores the close neighborhood of thegenetic material of both parents. If the two parents have a good fitness, onecan expect that the child will yield a similar or better fitness as only a smalldisruption is introduced in the original genetic material.
Once we carried out an initial study on the behavior of the crossover oper-ators introduced in this article, we carried out an experiment to select the bestcrossover operator for the problem from the ones proposed in this work andsome classic and well-known crossover operators. With that goal in mind, wedevise the following experiment: • We ranged the number of students from 50 to 150 with steps of 50, andwe ranged the number of supervisors from 5 to 30 with steps of 5. All27f the student and supervisor profiles were selected from the collecteddataset. The minimum workload of supervisors c j,min was set to 1 student(i.e., a supervisor will advise at least one student) and the upper boundsupervision quota c j,max of each supervisor was generated from a uniformdistribution U (4 , × × × • The mutation operator was set with a mutation ratio p mt = 0 .
05 and theprobability of carrying out a swap operation in a gene to be mutated wasset to p sw = 0 .
2. These values were found to be one of the best performingin the first experiment. • We tested the performance of the
Hopcroft-Karp , the greedy structuralpreservation , the uniform [26], and the [27] crossover operators. • The rest of the parameters were adjusted in the same way as defined inthe experiment carried out to optimize the mutation operator.Similarly to the first experiment, we employed the S-metric, the best fitnessfound for the students, and the best fitness found for the supervisors as metricsto assess the quality of the different configurations. The results of this experi-ment can be observed in Table 1. This table contains 4 sub-tables that describethe performance of the different crossover operators on the problem set.The first subtable in Table 1 summarizes the performance of the genetic al-gorithm configured with the proposed crossover operators plus the uniform and8-point crossover operator. At a first glance, one can observe that the greedystructural preservation operator tends to outperform the rest of crossover op-erators for all of the metrics. A one-sided Mann-Whitney test comparing theperformance of the aforementioned crossover operator with the individual per-formance of each of the other three crossover operators was carried out to assessthe statistical significance of the results. The test suggests that the greedy struc-tural preservation operator outperforms the rest of the crossover operators forthe S-Metric (i.e., the quality of the estimated Pareto frontier, to be minimized),and the best utility found for the supervisors. With regards to the best util-ity found for the students, the greedy structural preservation operator was alsothe best performing operator, although this time we could not find statisticaldifferences with the uniform crossover. Another interesting point that shouldbe raised is that the Hopcroft-Karp crossover tends to be amongst the worstperforming operators from the set. Despite the similarities between the greedystructural preservation and the
Hopcroft-Karp operator, the results suggest that This value of k was found to perform the best for the problems at hand α = 0 .
05, with the alpha values adjusted with the Bonferroni-Holm correction opcroft Greedy Uniform 8-pointS-Metric Best F. Stu
Best F. Sup.
S-Metric Hopcroft Greedy Uniform 8-point |S| = 50 |S| = 100 |S| = 150
F. Stu Hopcroft Greedy Uniform 8-point |S| = 50 |S| = 100 |S| = 150
F. Sup Hopcroft Greedy Uniform 8-point |S| = 50 |S| = 100 |S| = 150
Table 1: The performance of the crossover operators on the S-Metric, the best fitness foundfor the students, and the best fitness found for the supervisors over all of the problem sets(top), and detailed over different problem sets (middle and bottom). the ratio of new genetic material introduced by the greedy structural preserva-tion crossover is beneficial for the problem at hand. In addition to this, thetemporal complexity of the operator is lower than the
Hopcroft-Karp operator,making it more appropriate for this problem.The other three subtables offer a more detailed view on the performance ofthe crossover operators with problems of different size. Each cell represents theperformance of a given crossover operators with the problems of a given size.The performance is summarized in the form of the average over the differentproblems of that size, and the percentage of the problems of that size for whichthe crossover operator outperform the other operators. A closer look at thethree subtables suggests that for the smaller problem instances (i.e., 50 stu-dents) the four crossover operators tends to perform similarly. As the problemsize increases, so does the difference between the greedy structural preservation crossover and the rest of the operators. For the larger problem instances, theproposed crossover operator is the best performing operator for all of the met-rics. It should be highlighted that this is particularly true for the S-Metricand the best fitness found for the supervisors, as the operator was found tooutperform the other three for 89% and 99% of the cases respectively. This in-dicates that the greedy structural preservation operator is more suited for largerproblem instances, making it the best choice overall from the studied set.
In the previous subsection we have studied the individual performance ofeach of the configurable components of the GA. These studies only aimed atselecting the best possible configuration, but they did not focus on studyingwhether or not obtained solutions could be considered as good for the problem29 . Students F. SupervisorsSupervision capacity165
Supervision capacity172
Supervision capacity180
Table 2: Average percentage of optimality obtained by the proposed genetic algorithm for thebest fitness of the students, and the best fitness of the supervisors at hand. In this section we focus on comparing the quality of the solutions foundwith the optimal solution found by global optimal optimization methods. Morespecifically, we analyze the optimality of the best fitness found for the students,and the optimality of the best fitness found for the supervisors in the geneticalgorithm. In this experiment, we focus on the largest problem instances, asmetaheuristics tend to degrade their performance with the size of the problem.More specifically, the experiments were designed as follows: • The number of students was set at 150, and the number of supervisorswas set at 30. All of the student and supervisor profiles were selectedfrom the collected dataset. The minimum workload of supervisors c j,min was set to 1 student (i.e., a supervisor will advise at least one student)and the upper bound supervision quota c j,max of each supervisor wasgenerated from a uniform distribution U (4 , × • We selected the greedy structural preservation as the crossover operatorfor the GA. • The stop criteria was changed to keep running iterations in the GA unlessthe S-Metric of the estimated Pareto optimal frontier has not improved in20 iterations. At that point, we consider that the GA has converged. • Given a particular problem instance, the best possible solution for thestudents and for the supervisors were calculated executing two differentoptimization problems on
BARON [28].
BARON is a state-of-the-artglobal non-convex optimization algorithm that supports constrained andpure integer optimization problems. • The rest of the parameters were adjusted in the same way as defined inthe previous experiment.The main results of this experiment can be found in Table 2. The tabledepicts the average percentage of optimality obtained for the best solution foundfor the students, and the best solution found for the supervisors by the GA. Asit can be observed, the average percentage of optimality of the best solution forthe students ranges from 88 to 89% of the best fitness, while it ranges from 9330 igure 10: Convergence of the GA for an optimization problem with 180 students and 30supervisors. The left graph shows the convergence of the best solution found for the students,the middle graph depicts the convergence of the best solution found for the supervisors,while the right graph shows the convergence of the S-Metric for the estimated Pareto optimalfrontier. to 94% for the best solution for the supervisors. These results indicate that thePareto optimal frontier obtained by the GA contains solutions that are closeto both the optimal solution for the students and the optimal solution for thesupervisors. The gradual convergence of the GA for a particular problem canbe observed in Figure 10. As it is observable, the initial population of the GAis far from the optimal solutions (i.e., optimal solution for the students, optimalsolution for the supervisors, and the distance to the reference point (1,1) in theS-Metric). As several iterations are undertaken, the GA gradually convergestowards solutions that are closer to the optimal values.Moreover, it should be highlighted that the Python implementation of theGA obtained these estimations in an average of 247 seconds, while
BARON took approximately 1020 seconds per non-linear optimization problem and onlyobtaining a single solution each time. Not only the estimated frontier containssolutions that are close to the optimal one for both the students and the super-visors, but these are obtained in a reasonable amount of time compared to theexact method. Nevertheless, it should be considered that our approach aims forobtaining a Pareto optimal frontier, and the exact method computes a singlesolution. The former is preferred from the point of view of a human decisionmaker with possibly uncertain preferences. In addition, the GA was capable ofproviding an average of 27 solutions in the estimated Pareto optimal frontier,which also provides with diversity to the human decision maker.
6. Conclusions
In this article we have proposed a multiobjective genetic approach for thestudent-supervisor allocation. This optimization problem is a subclass of thestudent-project allocation problem. Given the hardness of the matching prob-lem, we have opted for a metaheuristic approach with that ability to take mul-tiple objectives into consideration. More specifically, we take into considerationthe students’ preferences with regards to research/project topics, as well as thelecturers’ preferences with regards to topics, which does not require the mas-sive proposal of projects prior to the allocation, and it avoids providing explicit31references on students as that may be regarded as a discriminatory practice.Furthermore, the genetic algorithms takes into consideration the constraints ofthe department in the form of lower and upper supervision quotas for lecturers,and attempts to provide a balanced workload allocation for lecturers.For this purpose, we have taken a Pareto optimal genetic scheme that aimsto provide human decision makers with trade-off opportunities. The geneticalgorithm employs a new mutation operator that can offer either explore thestructure of the allocation (i.e., the number of students supervised by each lec-turer) and the allocation itself. In addition, two new crossover operators havebeen specifically designed for the student-supervisor allocation problem: the
Hopcroft-Karp crossover operator, and the greedy structural preservation oper-ator. Both aim to preserve the allocation structure of one of the parents, thedifference being that the
Hopcroft-Karp crossover preserves also the original ge-netic material from parents, while the greedy structural preservation crossovermay introduce new genetic material. The theoretical and empirical complexityof both operators has been studied, with the complexity of the former operatorbeing linearithmic, and the complexity of the latter being linear. The geneticalgorithm has been tested with real data collected from the student-supervisorallocation process at Coventry University. The results show that (i) the muta-tion operator benefits from giving more importance to exploring the structureof the allocation; (ii) the greedy structural preservation operator outperformsclassic crossover operators for the problem at hand; (iii) and that the geneticalgorithm is capable of providing solutions that are very close to the optimalsolutions in a limited span of time, even for large problem instances.
Acknowledgements
This work is partially supported by funds of the Faculty of Engineering andComputing at Coventry University, and funds from EU ICT-20-2015 ProjectSlideWiki granted by the European Commission.
ReferencesReferences [1] A. A. Anwar, A. Bahaj, Student project allocation using integer program-ming, IEEE T. Educ. 46 (3) (2003) 359–367.[2] P. R. Harper, V. de Senna, I. T. Vieira, A. K. Shahani, A genetic algorithmfor the project assignment problem, Comput. Oper. Res. 32 (5) (2005)1255–1265.[3] D. J. Abraham, R. W. Irving, D. F. Manlove, Two algorithms for thestudent-project allocation problem, Journal of Discrete Algorithms 5 (1)(2007) 73–90. 324] D. F. Manlove, G. O’Malley, Student-project allocation with preferencesover projects, Journal of Discrete Algorithms 6 (4) (2008) 553–560.[5] D. Srinivasan, L. Rachmawati, Efficient fuzzy evolutionary algorithm-basedapproach for solving the student project allocation problem, IEEE T. Educ.51 (4) (2008) 439–447.[6] L. Pan, S. Chu, G. Han, J. Z. Huang, Multi-criteria student project allo-cation: A case study of goal programming formulation with dss implemen-tation, in: 8th International Symposium on Operations Research and ItsApplications, 2009, pp. 75–82.[7] M. M. El-Sherbiny, Y. M. Ibrahim, An artificial immune algorithm withalternative mutation methods: applied to the student project assignmentproblem, in: International conference on innovation and information man-agement (ICIIM2012), Chengdu, China, 2012.[8] K. Iwama, S. Miyazaki, H. Yanagisawa, Improved approximation boundsfor the student-project allocation problem with preferences over projects,Journal of Discrete Algorithms 13 (2012) 59–66.[9] A. Kwanashie, R. W. Irving, D. F. Manlove, C. T. Sng, Profile-based opti-mal matchings in the student/project allocation problem, in: InternationalWorkshop on Combinatorial Algorithms, Springer, 2014, pp. 213–225.[10] A. Kwanashie, Efficient algorithms for optimal matching problems underpreferences, Ph.D. thesis, University of Glasgow (2015).[11] H. O. Salami, E. Y. Mamman, A genetic algorithm for allocating project su-pervisors to students, Int. J. of Intelligent Systems and Applications 8 (10)(2016) 51.[12] F. Cooper, D. Manlove, A 3/2-approximation algorithm for the student-project allocation problem, in: LIPIcs-Leibniz International Proceedings inInformatics, Vol. 103, Schloss Dagstuhl-Leibniz-Zentrum fuer Informatik,2018.[13] M. Chiarandini, R. Fagerberg, S. Gualandi, Handling preferences instudent-project allocation, Annals of Operations Research (2018) 1–40.[14] P. Bir´o, T. Fleiner, R. W. Irving, D. F. Manlove, The college admissionsproblem with lower and common quotas, Theor. Comput. Sci. 411 (34-36)(2010) 3136–3153.[15] K. Hamada, K. Iwama, S. Miyazaki, The hospitals/residents problem withquota lower bounds, in: European Symposium on Algorithms, Springer,2011, pp. 180–191.[16] K. Hamada, K. Iwama, S. Miyazaki, The hospitals/residents problem withlower quotas, Algorithmica 74 (1) (2016) 440–465.3317] J. E. Beasley, Or-library: distributing test problems by electronic mail, J.Oper. Res. Soc. 41 (11) (1990) 1069–1072.[18] R. Aydo˘gan, P. Yolum, Learning consumer preferences using semantic sim-ilarity, in: Proceedings of the 6th International Joint Conference on Au-tonomous Agents and Multiagent Systems, AAMAS ’07, ACM, New York,NY, USA, 2007, pp. 1293–1300. doi:10.1145/1329125.1329401 .URL http://doi.acm.org/10.1145/1329125.1329401http://doi.acm.org/10.1145/1329125.1329401