Discrete Applied Mathematics

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1 Discrete Appied Mathematics 159 (2011) Contents ists avaiabe at ScienceDirect Discrete Appied Mathematics journa homepage: A direct barter mode for course add/drop process Ai Haydar Özer, Can Özturan Department of Computer Engineering, Boğaziçi University, 34342, Bebek, Istanbu, Turkey a r t i c e i n f o a b s t r a c t Artice history: Received 31 Juy 2009 Received in revised form 4 January 2011 Accepted 6 January 2011 Avaiabe onine 5 February 2011 Keywords: Add drop Student registration Timetabing Course scheduing Bartering Barter network Network fow Even though course timetabing and student scheduing probems have been studied extensivey, not much has been done for the optimization of student add/drop requests after the initia registration period. Add/drop registrations are usuay processed with a first come first served poicy. This, however, can introduce inefficiencies and dead-ocks resuting in add/drop requests that are not satisfied even though they can, in fact, be satisfied. We mode the course add/drop process as a direct bartering probem in which add/drop requests appear as bids. We formuate the resuting probem as an integer inear program. We show that our probem can be soved poynomiay as a minimum cost fow network probem. In our mode, we aso introduce a two-eve weighting system that enabes students to express priorities among their requests. We demonstrate improvement in the satisfaction of students over the currenty used mode and aso the fast performance of our agorithms on various test cases based on rea-ife registration data of our university Esevier B.V. A rights reserved. 1. Introduction In universities, course timetabing (CT), student scheduing (SS) and add/drop processes invove the coordination of various resources and entities. CT basicay deas with the aocation of time sots and cassrooms to courses by taking into consideration issues such as preferences of instructors and cassroom ocations. Given a timetabe, in SS phase, students seect courses according to their needs and preferences. Because of course and section quota restrictions or enroment baancing requirements among the sections, it is not possibe to satisfy the needs and preferences of a the students. Therefore, some poicy or agorithm needs to be empoyed in SS phase for the assignment of students to courses and sections. During the add/drop phase, a readjustment of the assignment soution in SS phase basicay takes pace by the addition, dropping and swapping of courses and/or sections. In the iterature, phases CT and SS have been extensivey studied (see, for exampe, surveys [6,7,20,27]). Some approaches tacked either CT or SS excusivey. Some approaches couped these two phases and soved the combined course timetabing and student scheduing probem. In this paper, our focus wi be on the add/drop process. Not much has been done for this phase we are aware of ony one work (that of Graves et a. s [14]) that addresses the add/drop process. The add/drop process has an important difference from that of CT and SS. A student may have been aready assigned to a seat in a course or section from SS phase and he may want to swap (barter) this seat that he owns with another seat owned by other students in another course or section. Hence, one can say that whereas CT and SS phases can be modeed as an assignment probem, for add/drop process bartering is a more appropriate mode. We were motivated to deveop a direct barter mode for the add/drop process because of some probems we noticed during add/drop periods at our Boğaziçi University. Since 1998, web based onine registration system has been used for course registration [5]. Before the beginning of each semester, students are admitted to the system and are aowed to take courses if both prerequisites of the courses are satisfied and the quotas of the courses permit. The system works on a first Corresponding author. Te.: ; fax: E-mai addresses: ozeraih@boun.edu.tr, ahozer@gmai.com (A.H. Özer), ozturaca@boun.edu.tr (C. Özturan) X/$ see front matter 2011 Esevier B.V. A rights reserved. doi: /j.dam

2 A.H. Özer, C. Özturan / Discrete Appied Mathematics 159 (2011) Fig. 1. Exampe probem for iustrating add, drop, and barter bids. come first served (FCFS) poicy basis and at the beginning of each registration period, a race occurs among students for popuar courses. Generay, the quotas of the popuar courses are fied within the first few hours of onine registration period. After the registration period, the semester begins and during the first week of the semester, the students attend and evauate their courses. At the end of this week, add/drop period of one week begins and the students are aowed to change their courses and/or sections of their courses. Because of the FCFS basis of the system and the quota restrictions, when a student drops a course, he may not be abe to take it again. This situation forces a student who wants to change his course, to first try to add a new course, and then drop the od course. Athough this does not pose a probem if the quotas of the courses are not fu, it does pose a probem for the popuar courses. It is observed in Boğaziçi University student registration system that the current FCFS based system causes deadock situations, and hence reduces the tota satisfaction of students. Athough different impementations of FCFS approach exist in different registration systems, a FCFS based systems are prone to the same probem. For instance, in UniTime [21,29], which is an open source enterprise system for automated construction of course timetabes and student schedues, when a student wants to add a course which is not avaiabe, the student is assigned to the wait-ist of that course. Wait-ists are processed automaticay in FCFS manner as one seat becomes avaiabe for the corresponding course. Therefore, since a student who wants to change his course cannot be sure whether he woud be assigned to the new course, he woud not want to drop the course he has aready assigned unti he obtains a seat in the new course. Thus, this woud aso ead to the same probem. In order to increase the efficiency of add/drop process compared to the current FCFS based system of our university, a direct barter mode for the course add/drop process is proposed. The objective of the mode is to increase the tota satisfaction of students whie preserving fairness among them. For this purpose, aong with the usua add and drop requests, this mode aows students to barter the courses they want to drop for the courses they want to add. Students express their requests through submitting mutipe add, drop and barter bids and in each add and barter bid, they can decare a set of aternative courses to be added. Besides, in this mode, they can indicate reative priorities of their bids and the courses they want to register for. For instance, if a student prefers course A over course B, and course B over course C, he just decares A B C. Furthermore, students can request the same course or the same set of courses in mutipe bids and can aso decare restriction sets in which ony one course can be added to the schedue. In this paper, we contribute a forma deveopment of the mode. We present a network fow based agorithm that aows us to sove the probems in strongy poynomia time. We aso compare the soutions of our mode with that of the FCFS approach based on rea-word student registration data and present the performance of our agorithms on various tests. In the next section, we present an exampe with which we expain our mode for the course add/drop process. In Section 3, we formay define and formuate our mode using integer programming. Then, in Section 4 we present a minimum cost network fow soution of our probem and in Section 5, we present the experimenta resuts. A review of the reated iterature is given in Section 6. Finay, the paper is concuded in Section A motivationa exampe and the mode In this section we present an exampe scenario for add/drop process on which we expain our direct barter mode. Assume that during the registration period, students Ai, Mehmet, Ayşe and Ası have been registered for courses STS , SOC , ESC and SOC respectivey. Murat, on the other hand, has been registered for both STS and PSY Suppose that during the add/drop period, the students decare add, drop, and barter bids as shown in Fig. 1. Bids 1 5 are exampes of a barter bid. In a barter bid, the eft hand side of the arrow indicates the course to be dropped and the right hand side indicates the course to be added. A barter bid as the name suggests enforces the student to drop the course on the eft hand side if he adds the new course on the right hand side. For instance, in bid 3 Ayşe wants to drop ESC if she coud add SOC to her course ist. Bids 6 and 7 are exampes of an add bid. An add bid states that the student wants to add the course on the right hand side without dropping any other course. Likewise, a drop bid, e.g. bid 8, states that the student wants to drop the course on the eft hand side without adding any other course. Bids 1, 4 and 6 are different from the others in terms of having a request set of more than one course on the right hand side. These bids are caed muti-bids. A muti-barter bid states that the student is indifferent, at east to some degree, to the

3 814 A.H. Özer, C. Özturan / Discrete Appied Mathematics 159 (2011) set of requested courses and he is wiing to drop the course on the eft hand side if he coud add any one of the courses in this set. Simiary, a muti-add bid states that the student wants to add any one of the courses in this set without dropping any other course. Since drop bids are not restricted with quota constraints, they shoud aways be satisfied. Therefore, we do not need to expicity incorporate muti-drop bids in the mode. Muti-bids can be considered as combinations of two or more singe bids which are XOR ed. For instance, in bid 1, Ai wants to add either PSY or STS but not both and drop STS on the condition that his add request is satisfied. This bid can be represented as a combination of two XOR ed bids, STS {PSY } and STS {STS }. Bid 1 is satisfied if exacty one of these bids is satisfied. By introducing muti-barter bids, without osing generaity, we can now safey assume that there are no two barter bids of a student that have the same course on the eft hand side since such bids can be combined and represented as one muti-barter bid. In addition to the muti-barter bid mechanism, the mode aows a student to mark a barter bid (either singe or muti-barter) as drop-uness-barter meaning that the student wants to drop the course on the eft hand side if bartering of this course for another course in the request set is not possibe. In the given exampe, it is indicated using a star above the arrow of the bid 5. In this bid, Murat wants to barter PSY for ESC and if ESC cannot be added, Murat wants to drop PSY Again, by further introducing drop-uness-barter mechanism, drop and barter requests can be combined and again without osing generaity, we can state that there cannot be any two bids of a student that have the same course on the eft hand side. The add/drop process based on the direct barter mode is a batch process and consists of two phases, a bid submission phase in which the students are aowed to submit bids to the system or retract bids from the system and a soution phase in which the optimum soution is cacuated. Depending on the duration of the add/drop period, these phases can be repeated as many times as necessary. For instance, for each day of the add/drop period, the bids can be coected from the students throughout the day and the soution can be cacuated at the end of the day and announced to the students afterwards. Expressing preferences weighted mode Athough the described unweighted mode heps to increase the tota satisfaction of students, it can further be improved by assigning weights to the bids. In the weighted mode, add, drop, and barter bids are defined respectivey as foows: w i {(ai1, w i1 ), (a i2, w i2 ),..., (a ip, w ip )} c w i d i d i {(c, 0)} w i {(ai1, w i1 ), (a i2, w i2 ),..., (a ip, w ip )} where i is the index of the bid, w i is the weight of the bid i, and d i is the course to be dropped. c denotes the nu course used for representing the course to be dropped or added for add and drop bids respectivey. In the weighted mode, weights are assigned not ony to the bids, but aso to the requested courses. Therefore, the request set contains tupes (a ij, w ij ) where a ij is the requested course, w ij is the associated weight, and p is the number of the requested courses. By assigning weights to the bids and the requested courses, the mode becomes more powerfu in the sense that it enabes students to express their preferences inside the bids. For instance, weight vaues can be assigned to a student s bids indicating the degree of his preferences for his bids. The weight of the most favored bid woud be the highest and the east favored woud be the owest. Likewise, for each muti-bid, weight vaues can aso be assigned to the requested courses in the request set if he is not totay indifferent to these courses. Considering the quota restrictions and the submitted bids, among the possibe courses in the request set, the one with the highest weight woud be added to the student s course ist. Besides the abiity to express preferences among the bids and the requested courses, the weighted mode aso enabes favoring some students over others. Specia students such as graduating students can aso be favored officiay by their department by increasing or maximizing the weights of their bids. This wi ensure that if the quotas of the courses are avaiabe then these students wi be the first to add the courses they want. Simiary, using the same mechanism, successfu students, i.e. the students with higher grade point average (GPA), can aso be favored depending on the poicy of the university. 3. Formuation of the mode The weighted direct barter mode is formay defined as foows: et C = {c 1, c 2,..., c m } be the set of m courses and Q = (q c1, q c2,..., q cm ) be the tupe of remaining quotas where q ck is the remaining quota of course c k (1 k m, q ck Z + {0}). Let S = {s 1, s 2,..., s t } be the set of t students. We define B as the set of bids submitted by a student s and the set of a bids, B, is defined as B = t =1 B. Each bid is denoted by a tripet, b i = (d i, w i, R i ), where d i is the course to be dropped for barter and drop bids or the nu course, c, for add bids (d i C {c }), w i R + is the weight and R i is the request set of the bid b i. The request set of a bid is either {(c, 0)} for drop bids or a set of two tupes, R i = {(a i1, w i1 ), (a i2, w i2 ),..., (a ip, w ip )}, for barter and add bids. Each tupe (a ij, w ij ) in R i indicates the requested course, that is the course to be added, and the associated weight respectivey (a ij C, w ij R + ). Finay, the set D B denotes the bids which are marked as drop-uness-barter. A bid b i is caed satisfiabe if at east one of the courses in the request set has one or more remaining quota or there exists at east one satisfiabe bid whose course to be dropped is in the request set of b i. Formay, given b i = (d i, w i, R i ) and b = (d, w, R ) the foowing predicate is true: i (satisfiabe (b i ) j ((a ij, w ij ) R i ((q aij Q q aij > 0) (satisfiabe (b ) d = a ij )))).

4 A.H. Özer, C. Özturan / Discrete Appied Mathematics 159 (2011) By the definition, a drop bids are satisfiabe. The objective of the mode is to find the set of satisfiabe bids that maximizes the sum of the weights, that is the sum of both bid weights and weights of the requested courses, and hence the tota satisfaction of students. In order to formuate the mode using integer programming, two binary variabes are introduced. The binary decision variabe x determines the satisfied bids and y determines the requested course which is added if the corresponding bid is satisfied. Formay, 1, x i = if bid i is satisfied 0, otherwise 1, and y ij = if course aij is added for bid i 0, otherwise. It shoud be noted that for a drop-uness-barter bid b i, the meaning of satisfaction is sighty different. x i = 0 means that the student drops the course without adding any other course and x i = 1 means that the student barters the course for another course. The integer programming formuation of the mode is as foows: maximize w i x i + α w ij y ij i b i B j (a ij,w ij ) R i (1) subject to x i y ij q ck + 1 ( k c k C) i b i (B\D) d i =c k i,j b i B (a ij,w ij ) R i a ij =c k i b i D d i =c k (2) x i j (a ij,w ij ) R i y ij = 0 ( i b i B R i {(c, 0)}) (3) x i = 1 ( i b i B R i = {(c, 0)}) (4) i,j b i B (a ij,w ij ) R i a ij =c k y ij 1 ( k, c k C s S) (5) x i, y ij {0, 1} ( i, j). (6) Note that in this formuation, the objective ine in Eq. (1) maximizes the sum of the bid weights and the weights of the courses in the request sets. In this equation, α factor is a constant positive number to be determined according to the actua weight vaues and the number of bids which is described in the next section. Eq. (2) enforces quota restrictions of the courses. For each course, the number of students dropping the course (incuding the drop and drop-uness-barter bids) pus the remaining quota of the course shoud be greater than or equa to the number of students who added the course. Eq. (3) expresses the satisfaction criterion: an add bid or a barter bid is satisfied if exacty one of the courses in the request set is added. Eq. (4) ensures that a the drop bids are satisfied. Finay, Eq. (5) prevents students from adding the same course to their schedue more than once Determining the weight vaues and the α factor The main objective of the direct barter mode is to increase the tota satisfaction of students whie preserving fairness among them. In this section, we propose a method for defining the parameters of the mode, that are the weights of the bids, w i, the weights of the requested courses, w ij, and the α factor, in accordance with this objective. In this method, each student, s, is responsibe for ranking his bids according to his preference instead of defining the actua weights of the bids. Based on this ranking, he constructs his preference ist, a permutation of his bids sorted in descending order of his preference. Then, this preference ist is used as B, the set of bids submitted by the student s. So, in the set B = {b (1), b (2),..., b (u) }, the bid b (1) is the most preferred bid with highest rank number of 1 and the bid b (u) is the east preferred bid with the owest rank number of u (i.e. : b (1) b (2)... b (u) ). Note that, for this definition we use a different indexing scheme for referring the bids in the set B in order to prevent confusion with the indexing scheme for referring the bids in the bid set B. Subscript of a bid denotes the owner of the bid and the superscript, which is the index number in the set B, denotes the rank of the bid. We aso denote the weight of a bid b (r) bids B of the student s, the weight of a bid b (r) is defined as foows: w (r) = 2 hr (, r b (r) B ). with w (r). Given the ordered set of In this function, h is the index of the owest ranked bid among a the bids, and therefore the minimum bid weight vaue, w min is 1. This function ensures that the weights of the bids that have the same rank among the students are equa and for each bid b (r) B, the weight of the bid is greater than the sum of the weights of the ower ranked bids in the same set. Therefore, this bid weight function enabes the mode to satisfy as many higher ranked bids as possibe whie preserving fairness among the students. (7)

5 816 A.H. Özer, C. Özturan / Discrete Appied Mathematics 159 (2011) Defining the weight vaues for the requested courses is straightforward. As for the bids, each student decares the courses in the request set such that the more preferred course comes before the ess preferred course (i.e. i b i B : a i1 a i2 a ip ). Since for each bid at most one requested course can be added to the students schedue in the optimum soution, the ony requirement for weights of the requested courses is to ensure that the more preferred requested course has higher weight vaue than that of the ess preferred course. Therefore, the weight vaue of a requested course is simpy defined as: w ij = m j + 1 ( i b i B) (8) where m is the number of courses (m = C ). This simpe function ensures that the requested courses with the same rank have equa positive weights among a the bids. As seen from the objective function given in Eq. (1), there are two objectives of the mode: the first objective is to maximize the number of satisfied bids according to bid weight vaues and the second objective is for each satisfied bid to add one requested course with the maximum possibe weight. In order to maximize the tota satisfaction of students, the first objective is favored against the second objective so that when finding the optimum soution among the feasibe soutions, the soution with the maximum sum of the bid weights is chosen as the optimum. However, if there are mutipe soutions with the same maximum sum of the bid weights, then the soution with the maximum sum of the weights of the requested courses is chosen among these soutions. In order to provide this feature, the ranges of these two types of weights shoud be separated in order to cance the effects of the atter to the former. For this purpose, a constant factor α for scaing the sum of the weights of the requested courses is introduced in the objective function. The α factor is defined as foows: α = 1 ( B 1) m. (9) The foowing proposition proves that using these weight functions and the α vaue, the first objective of the mode is favored against the second objective. Proposition 1. Given any two different feasibe soutions with different sums of the bid weights for the direct barter mode, the soution with higher sum of the bid weights has higher objective vaue according to Eq. (1) independent of weights of the requested courses. Proof. Let B 1 and B 2 be two sets of satisfiabe bids that correspond to any two feasibe soutions for the direct barter mode, and z 1 and z 2 be the corresponding objective vaues such that z 1 = w i + α and z2 = w i + α. (10) i b i B 1 j (a ij,w ij ) R i y ij =1 w ij i b i B 2 j (a ij,w ij ) R i y ij =1 We wi show that if the sum of the bid weights of the B 1 is greater than the sum of the bid weights of the B 2, then the objective vaue z 1 is aways greater than the objective vaue z 2. Therefore, we wi be proving that for any two feasibe soutions, the soution with the higher sum of the bid weights has higher objective vaue independent of the sum of the weights of the requested courses. Suppose that the sum of the bid weights of the B 1 is greater than the sum of the bid weights of the B 2, w i > w i. (11) i b i B 1 i b i B 2 The difference between the sums of the bid weights of B 1 and B 2 is w i w i = 2 k (k Z + {0}). (12) i b i B 1 i b i B 2 Then, the difference between the objective vaues z 1 and z 2 is z 1 z 2 = 2 k + α w ij i b i B 1 j (a ij,w ij ) R i y ij =1 i b i B 2 j (a ij,w ij ) R i y ij =1 w ij w ij. (13) Since exacty one requested course is added for each satisfied bid, the ower bound for the sum of the requested course weights of B 1 is w ij min w ij = 1. (14) i,j i b i B 1 j (a ij,w ij ) R i y ij =1

6 A.H. Özer, C. Özturan / Discrete Appied Mathematics 159 (2011) Because of the conditiona proof assumption in Eq. (11), B 1 cannot be a subset of B 2, and therefore the upper bound for the sum of the requested course weights of B 2 is w ij ( B 1) max w ij = ( B 1) m. (15) i,j i b i B 2 j (a ij,w ij ) R i y ij =1 Therefore, i b i B 1 j (a ij,w ij ) R i y ij =1 w ij i b i B 2 j (a ij,w ij ) R i y ij =1 Using α = 1/( B 1) m, the smaest difference between the objective vaues is: w ij 1 ( B 1) m. (16) z 1 z [1 ( B 1) m] (17) ( B 1) m 1 z 1 z 2 ( B 1) m z 1 z 2 0. Therefore, the soution with the higher sum of the bid weights has higher objective vaue independent of the sum of the weights of the requested courses. In genera, as the number of courses with remaining quotas increases, the number of soutions with identica vaues in the first summand of the objective function is ikey to increase. The reason is that for satisfiabe bids there wi be more than one aternative requested course that can be added. Hence, the weight mechanism for the requested courses and α factor mechanism wi pay an important roe for increasing the satisfaction of students in these cases by enabing their favored courses to be added to their schedue. (18) (19) 4. Soution procedure Since the direct barter mode can be formuated using integer programming, its probem instances can be soved using genera purpose integer programming sovers. However, resembance of this mode to the used car saesman probem (UCSP) in [22] and the poynomia time barter modes in [23,24] motivated us to search for a network fow based soution. Because of the bid weights and the recursive definition of bid satisfiabiity that causes circuar patterns in the soution ike UCSP, we modeed the direct barter probem as a minimum cost fow probem [1]. The minimum cost fow probem is defined as foows: et N(V, A,, u, c, b) denote a network with node set V, arc set A, ower bound (v, w), capacity u(v, w), cost c(v, w) vaues for each arc (v, w) A, and suppy/demand vaues b(v) for each node v V. Let x(v, w) represent the fow on arc (v, w) A. The minimum cost fow probem is defined as foows: Minimize c(v, w) x(v, w) (20) s.t. w (v,w) A v,w (v,w) A x(v, w) w (w,v) A x(w, v) = b(v) ( v v V) (21) (v, w) x(v, w) u(v, w) ( v, w (v, w) A) (22) where v v V b(v) = 0. To hep us in defining the network in our probem formay, we first introduce a set P, caed restriction-pairs set, which consists of course student pairs. The set P is defined as foows: P = {(c k, s ) c k C s S ( i, i, j, j i i b i, b i B (c k, w ij ) R i (c k, w i j ) R i )}. Thus, each pair (c k, s ) in P indicates that the student s requests the course c k in his at east two different bids, and therefore the student s must be prevented from adding the course c k more than once in the fina soution. Based on this definition, the minimum cost fow network can be constructed as foows: The set of nodes, V, consists of four types of nodes: (i) a course node c k for each course c k C, (ii) a specia node CENTER that represents the nu course to be dropped for add bids, (iii) a bid node b i for each bid b i B, (iv) a restriction node r k for each (c k, s ) P for preventing the student s from adding the course c k more than once.

7 818 A.H. Özer, C. Özturan / Discrete Appied Mathematics 159 (2011) Fig. 2. Minimum cost fow network of the exampe given in Section 2 with (capacity, cost) vaues on the arcs. The soution is shown with the bod arcs where one unit of fow passes in each bod arc. The set of arcs, A, consists of seven types of arcs: (i) an arc (c k, CENTER) for each course c k C with capacity equa to q ck and cost equa to 0 which represents the remaining quota of the course c k, (ii) an arc (CENTER, c k ) for each course c k C with capacity equa to + and cost equa to ϵ, (iii) an arc (d i, b i ) for each barter and drop-uness-barter bid b i = (d i, w i, R i ) with capacity equa to 1 and cost equa to w i, (iv) an arc (CENTER, b i ) for each add bid b i = (c, w i, R i ) with capacity equa to 1 and cost equa to w i, (v) for each course c k C and for each student s S such that (c k, s ) P: (a) an arc (b i, r k ) for each bid b i = (d i, w i, R i ) B if there exists a tupe (c k, w ij ) R i with capacity equa to 1 and cost equa to α w ij, (i.e. i, j, k, s S c k C b i B (a ij, w ij ) R i c k = a ij (c k, s ) P: an arc (b i, r k )), (b) an arc (r k, c k ) with capacity equa to 1 and cost equa to 0, (i.e. k, s S c k C (c k, s ) P: an arc (r k, c k )), (vi) for each course c k C and for each student s S such that (c k, s ) P: (a) an arc (b i, c k ) for each bid b i = (d i, w i, R i ) B if there exists a tupe (c k, w ij ) R i with capacity equa to 1 and cost equa to α w ij, (i.e. i, j, k, s S c k C b i B (a ij, w ij ) R i c k = a ij (c k, s ) P: an arc (b i, c k )) (vii) an arc (b i, CENTER) for each drop-uness-barter bid b i D with capacity equa to 1 and cost equa to w i. Lower bounds (v, w) for a arcs (v, w) A are set to 0. Simiary, there is no suppy or demand for any node in the network, and therefore b(v) = 0 for every node v V. Note that ϵ which is used as the cost of the arcs of type (ii) is the smaest possibe positive number representabe on the computer. It is used to prevent zero cost cyces. The minimum cost fow network for the exampe probem given in Section 2 and its soution can be seen in Fig. 2. As stated earier, a drop bids shoud aways be satisfied and since they are aways part of the soution, they need not to be incuded in the network. Therefore, before constructing the network, as a preprocessing step a drop bids are marked as satisfied and the remaining quotas of the courses are increased accordingy. For instance, if there are z drop bids for the course c k, after satisfying these bids the remaining quota of the course becomes q ck + z. So, when presenting the network, we assume that there wi be no drop bids in the bid set B and the set of remaining quotas Q is adjusted accordingy. This simpe preprocessing step eiminates drop bids and reduces the network size. Therefore, for the exampe probem, the drop bid 8 is marked as satisfied beforehand and the quota of the course SOC is increased by one. Verifying the correctness of the described network is straightforward. The arcs of types (iii) and (iv) represent the binary decision variabes x i for barter and add bids respectivey. Therefore, additive inverses of the bid weights, w i, are used as the costs of these arcs. This statement is aso vaid for the drop-uness-barter bids on the condition that there is no fow on the corresponding arc of type (vii). However, if there is a fow passing through both the arc of type (iii) and the arc of type (vii) for a drop-uness-barter bid, this means that ony the drop part of the bid is satisfied. In this case, this bid is considered as unsatisfied in accordance with the IP formuation in Section 3 and the sum of the costs of the corresponding arcs of type (iii) and (vii) are zero. As in the case of the bid weights, additive inverses of the weights of the requested courses, α w ij, are used as the costs of type (v.a) and (vi.a) arcs that represent binary decision variabes y ij. Costs of the other arcs are zero. Therefore, minimization of the cost yieds maximization of the sum of the weights. In order to satisfy a bid, one unit of fow must fow from the node representing the course to be dropped (for add bids, from the center node) to the node representing

8 A.H. Özer, C. Özturan / Discrete Appied Mathematics 159 (2011) Fig. 3. Network for iustrating the usage of the restriction nodes for a course restriction set with (capacity, cost) vaues on the arcs. the course to be added. The capacity imits on type (iii) and (iv) arcs ensure that ony one of the requested courses is added for each satisfied bid. Aso, the capacity imits on type (v.b) arcs prevent students from registering for a course more than once. The arcs of type (i) represent the remaining quotas of the respective courses and type (ii) arcs aow satisfaction of barter bids when the courses to be dropped are not requested by any other satisfied bid. Finay, the arcs of type (vii) aow barter bids which are marked as drop-uness-barter to drop the course if the bartering is not possibe. Quota restrictions of the courses are enforced using the fow conservation property of the network nodes c k that correspond to the courses. Outgoing fow from a course node is restricted with the remaining quota (adjusted vaue according to the drop bids in the preprocessing step) pus the number of satisfied bids that drop the course. When the minimum cost fow is found on the network, winning bids can be determined by checking fow on arc types (iii) and (iv) for barter and add bids respectivey. If the fow on an arc of these types is 1, it shows that corresponding barter or add bid is satisfied and it is in the optimum soution. Simiary, the arc of types (v.a) or (vi.a) that originates from the winning bid determines the course to be added for that bid. Since there can ony be one arc with 1 unit of fow among these arcs, the head of this arc shows the course to be added for the winning bid. There are strongy poynomia agorithms for soving minimum cost network fow probems such as the minimum mean cyce-canceing agorithm with time compexity O( V 2 A 3 og V ) [13] and the enhanced capacity scaing agorithm with time compexity O(( A og V )( A + V og V )) [1]. Since the minimum cost fow network for a direct barter probem instance can be constructed in poynomia time using the above agorithm, the optimum soution of the direct barter probem can aso be found in poynomia time and hence the IP formuation given in Section 3 is in P. Extending the functionaity of the restriction nodes The function of the restriction nodes in the network is to prevent a student from registering the same course more than once. However, the function of these nodes can be extended so that instead of a singe course, any number of disjoint course restriction sets can be defined for each student such that the student can register for at most one of the courses in this set. This is especiay usefu when a student requests more than one section of the same course or a set of conficting courses in his at east two bids. For instance, assume that a student submits the foowing bids: CMPE w 1 {(CMPE ,w 11 ),(CMPE ,w 12 ),(CMPE ,w 13 )} CMPE w 2 {(CMPE ,w 21 ),(CMPE ,w 22 ),(CMPE ,w 23 )} It is cear that the student cannot register for two different sections of CMPE 230 at the same time. Therefore, in order enforce this restriction, we define a course restriction set that consists of restricted courses CMPE and Instead of using separate restriction nodes for these restricted courses, a spit restriction node pair is introduced for this set as shown in Fig. 3. Then, for each bid of the student that requests at east one of the courses in this set, an arc is drawn from the corresponding bid node to the first node of the pair. The capacity of this arc is set to 1 and the cost of this arc is set to the additive inverse of the weight of the highest ranked restricted course in the bid. Additionay, an arc is drawn between the pair nodes with one unit of capacity and zero cost. This arc imits the number of restricted courses to be added to one. Finay, for each restricted course, an arc is drawn from the second node of the pair to the corresponding course node with a capacity of 1 and cost of 0. This procedure is repeated by introducing a restriction node pair for each course restriction set defined. 5. Experimenta resuts In order to estimate the rea-word performance and the quaity of the soutions of our barter mode, we deveoped a test case generator based on rea-word student registration data obtained from the Boğaziçi University registration system [5] for the academic year Since the number of students in our university is reativey sma, that is 7095, we generated a statistica profie using the actua registration data and determined the parameters of the test case generator accordingy so that it is capabe of generating test cases for arbitrary number of students. The parameters of the test case generator and its source code can be found in [28].

9 820 A.H. Özer, C. Özturan / Discrete Appied Mathematics 159 (2011) Fig. 4. Graph of the number of students vs. running time (seconds) of the network sover. Tabe 1 Running times of the network sover for the test cases (seconds). # Students # Courses Avg. # Bids Running time (s) a , , , , , , , ,000 16, , a This row presents the resuts for Boğaziçi University. mean stdev We conducted two different experiments on a dedicated 64 bit Inte Xeon 2.66 GHz workstation with 8 GB memory using Linux operating system. We used CS2 software which contains a sover for the minimum cost fow probems based on scaing push reabe agorithm [8,12]. In the first experiment, a group of 20 test cases are generated for each seected number of students ranging between 7095 and 100,000. The average running time of the network sover for each group and the corresponding standard deviation are presented in Tabe 1 and the associated pot is depicted in Fig. 4. As seen from the resuts, the sover finds the soution of the probem instances with 100,000 students and approximatey 320,000 bids in ess than 21 s which is quite sma. For the case of our university, on the other hand, each instance is soved in ess than one second. In the second experiment, the soutions of our barter mode are compared with the currenty used FCFS based system. The purpose of this experiment is to present the improvement in the optimum soutions of the test cases over the FCFS approach under different occupancy rates for the courses. Thus, the importance of the introduced bartering mechanism and the weighting mechanism coud be observed. In order to simuate the FCFS system, a random permutation of the bids in each test case is generated by preserving the preferred order of bids of each student. The bids in the permuted ist are processed one by one, simuating the way the students submit the bids to the registration system. The processing step is straightforward; for each bid, the remaining quotas of the courses in the request set is checked in the order of students preferences and if one empty sot is found, the course is added to the schedue of the student. If the processed bid is a barter bid, then the course to be dropped is aso removed from his schedue and the remaining quota of that course is increased by one. The whoe simuation process is repeated up to five times for the unsatisfied bids. The second experiment is conducted for two different numbers of students, that is 7095 and 50,000, where the number of courses are 1158 and 8160 respectivey. For each number of students, the ratio of the courses without remaining quota to the number of a courses, caed p, is varied between 0.20 and 0.99 meaning that approximatey (100 p)% of the courses have no remaining quota. For each configuration, again a group of 20 test cases are generated. The resuts of this experiment are given in Tabe 2. For the test cases with 7095 students, assuming that 20% of the courses are fu, the number of satisfied bids in the soution found by the barter mode is approximatey 12% higher than that of the FCFS based system. For the case of our university where approximatey 28% of the courses are fu, the barter mode provides 15% better resuts. It is remarkabe that the improvement percentage for the number of satisfied bids increases exponentiay as p increases. In the extreme situation where the courses are 99% fu, approximatey 2800% improvement over the FCFS system is observed. It shoud aso be noted that in the barter mode the standard deviations are aso very sma reative to the mean vaues so that the quaity of the soutions found do not vary much. The resuts of the test cases with 50,000 students are aso very cose to the test cases with

10 A.H. Özer, C. Özturan / Discrete Appied Mathematics 159 (2011) Tabe 2 Improvements in the soutions of the barter mode over the FCFS mode. # Stu. (p) # Bids # Satisfied bids (mean/stdev) # Satisfied students (mean/stdev) ,000 FCFS Barter Impr. (%) FCFS Barter Impr. (%) ,757 18,367/148 20,642/ / / a ,725 17,823/176 20,409/ / / ,685 16,847/184 19,905/ / / ,761 14,224/260 18,656/ / / , /437 17,212/ / / , /497 16,300/ / / , /503 16,050/ / / , /123 15,569/ / / , ,495/ ,312/ ,285/53 48,040/ , ,538/ ,396/ ,469/68 48,033/ , ,492/ ,704/ ,625/167 48,039/ ,274 64,312/ ,217/ ,416/383 48,017/ ,052 35,624/ ,162/ ,203/691 47,675/ ,477 18,984/ ,228/ ,084/731 47,010/ , / ,539/ /476 46,300/ a This row presents the resuts for Boğaziçi University. Fig. 5. Graph comparing the number of satisfied students in the soutions of the barter mode and the FCFS mode students showing that the mode is scaabe to the universities with higher number of students without sacrificing the quaity of the soutions. By virtue of the weighting mechanism, the barter mode aso improves fairness among the students by increasing not ony the number of satisfied bids but aso the number of satisfied students. As seen from the pot in Fig. 5, the number of satisfied students in the barter mode whose at east one bid is satisfied is approximatey 96% of the tota number of students and decreases sighty to 93% as p goes to However, in the FCFS mode, starting from 95%, this ratio drops to 7%. 6. Previous work As stated earier, the CT and SS probems have been covered extensivey in the iterature. Some studies have addressed soey the CT probem; severa integer inear programming [3,9,18] modes and heuristic methods [3,9,15] have been proposed. For the SS probem, Reeves and Hickman [25] and Wioughby and Zappe [30] have proposed a mixed integer and network programming modes respectivey, and Avarez-Vades et a. [2] have appied Tabu search techniques. In [16,26], on the other hand, unified optimization modes that address both probems are presented. There are aso open source (for exampe [29]) and commercia packages (for exampe [17]) that address these probems. For further detais on these probems, the reader is referred to surveys by Burke and Petrovic [6], Carter and Laporte [7], Lewis [20] and Schaerf [27]. In the remaining of this section, we first review Graves et a. s work [14] in detai which has addressed the student course add/drop process as we do in this paper. Then, in Section 6.2, we discuss the reationship between our mode and a somewhat reated probem caed the stabe coege admission probem (SCAP) [10,4].

11 822 A.H. Özer, C. Özturan / Discrete Appied Mathematics 159 (2011) Reationship between Graves et a. s work and the barter mode Graves et a. [14] propose an auction based market approach compete with cearing prices for aocating course sections to students. Their mode consists of two rounds. In the first registration round, which is caed registration bidding system (RBS), students are granted bidding points (i.e. registration money) which they can use to bid on desired schedues. During this period, students are aowed to pace course seections as their bids together with the money they wi pay for each schedue. The bids are ranked in descending order of bidding points and are seected if requested course capacities are avaiabe. At the end of the registration period, the prices of the courses are determined. Each successfu bidder pays the sum of the prices of the assigned courses to him instead of the price he offered for his bid. Therefore, it is possibe that a successfu bidder may not have enough money in which case a subsidy given by the system covers the deficiency. Subsidies not paid back during add/drop phase as a resut of dropping courses are simpy forgotten (waived) by the system. Hence, we beieve that if this fact is known by the students, then it coud easiy be abused by offering high prices for their schedues. Since the winning bids can be subsidized and the subsidized amounts can be forgotten, this then introduces fairness probems in Graves et a. s approach. In the second round, which is caed drop/add/swap (DAS) round, Graves et a. introduce a course swapping idea. This corresponds to the barter scheme that we propose in this paper. As in the RBS round, an auction based approach is used. Students submit add, drop and swap bids together with the amounts of bidding points that are carried forward from the RBS round (if any). After the end of the round, a inear program whose objective is to maximize the sum of the bidding points of the satisfied bids is soved. By soving the inear program, the students are assigned to the courses and aso the prices of the courses in terms of bidding points (dua prices) are determined. As in the registration round, each student pays the actua amounts of his satisfied bids cacuated according to the determined prices of the courses. However, in this case it is stated that subsidies are not aowed. Since there is no forma treatment for this probem except for a simpe inear programming exampe, it is not cear how the restriction for preventing the subsidies is appied in their mode. This aso prevents one from impementing their mode. Athough, by the definition of dua prices, the actua amount to be paid for each satisfied bid is bounded by the offered price for that bid, this does not sove the subsidy probem since each student may submit more than one bid. Thus, a further constraint is necessary. In fact, the foowing proposition shows that introducing a budget constraint in order to prevent subsidies, i.e. the sum of the bidding points of the satisfied bids of a student shoud be ess than or equa to the amount of bidding points owned by that student, to the given inear programming exampe makes the resuting probem NP-hard. It shoud be noted that the forma definition of the DAS probem given beow is constructed by us according to the inear programming exampe and the expanations given in [14]. Proposition 2. The decision version of the DAS probem with budget constraint is NP-compete. Proof. Let Π be the decision version of the DAS probem with budget constraint. Π is defined as foows: given a set of courses, C = {c 1, c 2,..., c m }; a sequence of remaining quotas, Q = {q c1, q c2,..., q cm } where q ck is the remaining quota of course c k (1 k m, q ck Z + {0}); a set of students, S = {s 1, s 2,..., s t }; a set of bids, B = t =1 B where B is the set of bids of a student s (1 t) and each bid is denoted by a tripet, b i = (d i, a i, p i ), where d i is the course to be dropped for barter and drop bids or the nu course, c, for add bids (d i C {c }), a i is the course to be added for barter and add bids or the nu course, c, for drop bids (a i C {c }), and p i is the amount of bidding points offered by the student for bid b i (1 i n = B, p i Z + {0}); a set of bid restrictions, L = { 1, 2,..., z } that consists of mutuay disjoint subsets of bids, y B (1 y z), such that at most one of the bids in y can be satisfied (e.g. a student may put a restriction on two of his add bids so that ony one of them can be satisfied or the system may enforce a restriction on two barter bids of a student in which the same course is dropped); a sequence of bidding points owned by students, F = {f 1, f 2,..., f t } where f is the amount of bidding points owned by student s (1 t, f Z + {0}); a positive integer K ; is there a subset B B such that the foowing inequaities are satisfied? p i K (23) i b i B p i f ( s S) (24) i b i B a i =c k 1 i b i (B B ) 1 q ck ( k c k C) (25) i b i B d i =c k 1 1 ( y y L). (26) i b i ( y B )

12 A.H. Özer, C. Özturan / Discrete Appied Mathematics 159 (2011) In this formuation, Eq. (23) ensures that the sum of the offered bidding points for a satisfied bids (B ) is greater than or equa to the positive integer K. Eq. (24) is the budget constraint which prevents the sum of the bidding points of the satisfied bids of a student from exceeding the amount of bidding points owned by that student. The quota restrictions are enforced in Eq. (25). For each course, the number of students who drop the course pus the remaining quota of the course shoud be greater than or equa to the number of students who add the course. Finay, Eq. (26) ensures that bid restrictions are appied such that for a y (1 y z), at most one of the bids in y L is satisfied. If we have a certificate that consists of B B, this certificate can be verified in poynomia time by checking Eqs. (23) (26). Therefore Π is in NP. Next, we present a poynomia time transformation from the subset sum probem. Let Π be the subset sum probem (see for exampe: [19, p. 73] and [11, p. 247]) which is defined as foows: given a finite set U, a weight vaue w(u i ) Z + for each u i U (1 i U ), and positive integers C and K, is there a subset U U such that w(u i ) K (27) i u i U w(u i ) C. (28) i u i U Let Π (U, w(u i ), C, K) be an instance of the subset sum probem. It can be transformed to Π in poynomia time as foows: et the set of courses C consist of U courses (C = {c 1, c 2,..., c U }) and the remaining quotas of a the courses be 1 (q ck = 1, k = 1, 2,..., U ). The students set S consists of 1 student (S = {s 1 }) and for each u i U, the student s 1 submits an add bid b i requesting the course c i with a price of w(u i ) (i = 1, 2,..., U ). The amount of bidding points of the student s 1 is C (F = {C}). Let the set of bid restrictions, L, be empty. Since in each bid exacty one unique course is requested and the remaining quotas of a the courses are 1, Eq. (25) aways hods independent of the set B. Thus, for the transformed probem instances, Eqs. (23) (26) reduce to the inequaities of the subset sum probem: w(u i ) K (29) i b i B w(u i ) C (30) i b i B and therefore, the soution of a probem instance of Π is aso the soution of the corresponding probem instance of Π, and the soution of a probem instance of Π is aso the soution of the probem instance of Π. Since Π is in NP and the subset sum probem is NP-compete, the decision version of the DAS probem with dynamic credit constraint is NP-compete. Besides the subsidy and the associated unfairness probems in Graves et a. approach, students are aso responsibe for deciding the prices of the bids according to certain upper and ower bounds. However, determining the prices can be cumbersome for the students because of the combinatoria nature of the mode. Furthermore, since the remaining points of the students are transferred to the next semester, decision making wi become tougher since the students shoud aso consider the foowing semesters. Finay, we note that athough the students perception of the quaity of their schedues is not quantified, Graves et a. estimate one percent increase in the quaity of schedues using their mode Reationship with the stabe university admission probem The preference based ordering of bids and requested courses indicates a reationship between the direct barter mode and the stabe coege admission probem (SCAP) [10,4]. In the SCAP, there are two sets of agents, coeges and students. Each coege has strict preferences over the students and can accept a imited number of students. Each student, on the other hand, can enro to ony one coege and has aso strict preferences over the coeges. The SCAP is defined as finding a matching of students to coeges, caed a stabe aocation, such that no unmatched pair of opposite agents woud simutaneousy be better off if they were matched together. In genera, the SCAP cannot be used to sove our barter mode. However, if we consider a simpe specia case of our mode in which ony add bids are aowed and the weights of the bids are unique, then this simpe probem can be reduced to the SCAP as foows 1 : et the set of bids B and the set of courses C in our barter mode map to the row agents I (students), and the coumn agents J (coeges) in [4] respectivey. The function π(i, j) in the SCAP indicates whether student i wants to admit to coege j or not. Therefore, for each bid b i B and for each course c i C in our mode, if c i is requested in bid b i, then π(i, j) is set to 1. Otherwise, it is set to 0. For each bid b i B in our mode, the weights of the requested courses correspond to the strict coege preference of the student i. For each course c k C in our mode, the weights of the bids that request 1 Note that the atter requirement cannot be satisfied when the bid weight function in Eq. (7) is used. Therefore, even the instances of direct barter mode consisting of add bids cannot be reduced to the SCAP.