A methodology for projecting course demands in academic programs

Computers ind. Engng Vol. 12, No. 2, pp. 99-103, 1987 0360-8352/87 $3.00 + 0.00 Pergamon Journals Ltd Printed in Great Britain A M E T H O D O L O G Y FOR PROJECTING COURSE D E M A N D S IN ACADEMIC PROGRAMS BALASUBRAMANIAN RAM, SANJIV SARIN a n d ARUP MALLIK Department of Industrial Engineering, North Carolina A & T State University, Greensboro, NC 27411, U.S.A. (Received for publication 18 September 1986) Abstract--Several issues in academic course planning have received the attention of researchers. Foremost among them are problems involving faculty assignment to courses and course timetabling. This paper considers the problem of selecting courses and number of sections of each to offer based on demand projections. An algorithmic procedure is proposed to make this projection. 1. INTRODUCTION Several papers have appeared in literature in the area of academic course planning. The central issues associated with this problem are: (1) Which courses to offer? (2) Who is to teach each course? (3) When should each course be scheduled? Assignment of faculty to courses has been addressed by Andrew and Collins [1], Tillett [2], Breslaw [3], and McClure and Wells [4], [5]. The issue of course time-schedules is the well known timetabling problem and is discussed in Broder [6], Wood [7], Grimes [8], Leighton [9], Carter [10] and Lotfi and Sarin [11]. We are not aware of any work that addresses question (1) above. This paper presents an approach to answer this question. Sometime during the middle of a semester, the chairman of an academic department has to decide which courses and how many sections of each are to be offered for the next semester. This problem is particularly critical when the department does not have the resources to offer all its courses every semester. A good estimate of demand for the courses is essentially the basis for solving this problem. The common factors that affect demand for courses are listed below: (1) (2) (3) (4) (5) (6) (7) Enrolment in various academic programs. Enrolment in currently offered courses. Courses completed by students in various academic programs. Failure rates in various courses. Addition or deletion of academic programs. Changes in curricula of existing programs. Students transferring from another university or from a different academic program. We propose a simple methodology to estimate demand for courses and illustrate the use of the estimate in planning course offerings. The method addresses all the above factors. 2. THE METHOD In describing our method, we c o n f n e our attention to a single academic program and the courses related to it. The method is based on prerequisite and corequisite structure 99 100 BALASUBRAMANIAN RAM et al. for a curriculum. This structure may be represented as a network with courses represented as nodes. Figure 1 shows such a network for a typical academic program. Nodes B and F in the figure are the starting and finishing points respectively for the program. We can think of B as being a fictitious course which every freshman is assumed to have "passed" as soon as he or she joins the program. Similarly, F may be viewed as another fictitious course that will automatically be "passed" once all requirements for the program are completed. We define the following terms: (1) P(i): Set of prerequisite courses to course i. (2) C(i): Set of corequisite courses for course i. N / / / / / / / / / / \\ j J i .@ = ---e.- Prerequisite Corequisite Fig. 1. Prerequisite and corequisite structure. Projecting course demands 101 (3) E~: Number of students registered in course i during semester t. (4) S~: Number of students successfully completing course i in semester t. (5) T~: Cumulative number of students who have passed course i up to the end of semester t - 1 (since the inception of the program). (6) Pi: Fraction of students expected to pass course i in any semester. (7) D~: Number of students expected to register for course i in semester t. D~ represents our estimate of demand for a course. This demand is computed as the number of students who have successfully completed all prerequisites. The procedure also ensures that corequisite requirements are met. The following steps formalize the method. (1) Initialization step • Set t ~---0. • Initialize T~ for each i; E~ for all i, i #: F; and pi for i, i #: F. Set E~ ~ 0. • Define P(i) for all i, i :/: B; C(i) for all i, i :/: B, F. (2) Mid-semester step • Compute DI~ + 1 ~__. Minimum (T: + p/E:} - (T~ + piE~) j • P(i) for all i, i :/: B. • Compute X: +~ = Minimum {[T~ + p/E:+ DI: +~ - T~ - piE~ - DI~ + ~], 0} 1'• C(i) for all i, i ~ B. • Compute D: +~ = Dlti +1 + X : +1 for all i, i :/: B. (3) Beginning-of-next-semester step • • • • Input S~ for all i, i :/: F. Update p,. ~ (PiT~ + S~)/(T~ + E~) for all i, i~:B, F. S e t t ~-- t + l . Compute T~ = T~- x + S~- ~ for all i, i :/: F. T~ = Minimum {T~} j e P(F) • Update T~ for all i for all new transfer students (see example in the next section). • Input E~ for all i, i :/: F. Set E~ ~- 0. • Go to Step 2. The above steps require elaboration. In the initialization step, the prerequisite/ corequisite structure is input. T °, E ° and p; values must be obtained from student records. The main step is step 2 in which course demands for the subsequent semester are comPUted. The expression for computing DI[ +~ ensures that potential students for course i are those that have satisfied all its prerequisites. X[ +~ represents an adjustment to DI[ +1 to ensure that corequisite requirements are met. The expression for X~/÷l is based on the following condition that must be satisfied for corequisite courses. T ~ + E ~ + D I ~ t> ~ + E ~ + D l t i f o r a l l j • C(i). In case DI~ obtained in step 2 of the procedure violates the above inequalities, it is adjusted by X~ to obtain D~. Step 3 can be performed at the end of the semester when S~ values become available. The passing rates Pi can also be updated at this point. The computations for "courses" B and F require special handling. E~ denotes the number of students granted admission for semester t + 1, and S~ represents the number that actually join in semester t + 1; PB is defined similarly. Tt~ represents the cumulative number of students that have graduated, while D~:÷~ is the number of students that can potentially graduate in the following semester. BALASUBRAMANIAN RAM et al. 102 In the next section, we illustrate the application of the above method to a small example problem. 3. AN EXAMPLE Consider the example curriculum in Fig. 2, involving 7 courses. Table 1 summarizes the enroiment and passing counts over six consecutive semesters. The data is chosen to illustrate the salient features of the procedure. Exceptional cases such as transfer students have also been included (see semester 3). Notice that S~ ~< E~ for every i and t. We assume that Pi = 1 for all courses. In order to translate course demands into a decision on h o w many sections to offer, we use a rule summarized in Table 2. Fig. 2. N e t w o r k for e x a m p l e p r o b l e m . Table 1. Data for example problem t Eh S~ E~ S~ 0 1 2 3 4 5 6 40 25 42 27 39 30 49 . 26 19 30* 25 36 29 . 21 25 30 21 10 50 19 22 28 20 9 NA E~ . S~ E~ S~ 0 13 27 21 27 NA . 0 0 0 36 20 50 0 0 0 17 19 NA . 0 14 28 21 28 0 E~ . S~ E~ S~ 0 0 0 36 10 NA . 0 0 0 0 35 0 0 0 0 0 30 NA . (1 0 0 39 14 39 E~ S~ E~ S~ 18 26 22 35 40 30 17 25 19 31 39 NA 0 12 30 20 33 40 0 12 26 17 30 NA . * 1 Transfer student who has completed courses 1, 2 and 3. Table 2. Selection rule for number of sections DI Number of sections for semester t 0-14 15-54 55-94 95-134 0 1 2 3 Table 3. T',, E~, DI calculations Tl( i=B 0 1 2 3 4 5 6 6t 0,40,-26.25.-45,42,-75",27,-100,39,-136,30.-165,49,-135,49,-- i= 1 0, 0,40 0,21,30 19,25.43 42*,30,30 70,21,48 90,10,66 99,50,65 69,50.65 i=2 0, 0, 0 0, 0,18 0.14.29 14",28,22 41.21,29 62,28,10 89. 0,60 59, 0,60 i=3 0. 0, 0 0. 0, 0 0. 0,14 1", 0,41 1.36,25 18,20,52 37.50, 2 7,50, 2 t D~ t+ i E~, i=4 0, 0, 0 0, 0. 0 0, 0,14 0, 0,42 0,39,23 36.14,40 46,39, 4 16.39, 4 i=5 0, 0, 0 0, 0, 0 0, 0. 0 0, 0, 0 0, 0,37 0,35, 3 30, 0,55 0, 0,55 * Adjusted for 1 transfer student, t After adjusting T~ for graduated students. i=6 0, 0,40 0,18,33 17,26,44 42,22,38 61,35,43 92,40,34 131,30.53 101,30,53 i=7 i=F 0, 0, 0 0, 0,18 0,12,31 12,30,22 38,20,38 55,33,44 85.40,36 55,40,36 0,--.0 0,--,0 0,--,0 0,--,0 0,--,0 0.--,0 30,--,0 0,--,0 Projecting course demands 103 Tables 3 and 4 present results obtained upon using the proposed method. Note that the data for semester t in Table 1 results in the row corresponding to semester t in Table 3 which in turn translates to decisions for semester t + 1 shown in Table 4. For instance, D33 in Table 3 is 14. As a result, no section for course 3 is opened in semester 3 and therefore E33 is zero. In semester 3, we have a transfer student who has already completed courses 1, 2 and 3. T 3, T 3 and T 3 are incremented by 1 to acount for this. The method will work even if all T[ values are decremeted by Min {T~) = T~. This is illustrated in the last row of Table 3. i Table 4. No. of sections to bc offered Course i t i= 1 1 1 i=2 . 2 3 4 5 6 7 1 1 1 1 2 2 1 1 1 1 -2 i=3 . . --1 1 2 -- 4. i=4 i=5 . --1 1 1 -- ---1 -2 i=6 i=7 1 -- 1 1 1 1 1 1 1 1 1 1 1 1 A D D I T I O N A L NOTES The above approach can be extended to plan for campus-wide course offerings. Considering each academic curriculum, courses can be classified as (i) departmental (e.g. IE 102 in Fig. 1), (ii) non-departmental (e.g. Chem 101 in Fig. 1) and (iii) elective courses (not shown in Fig. 1). Departmental courses can be handled in a decentralized manner by directly using the proposed method. Demand for required non-departmental courses stems from students belonging to various academic programs. These individual demand values can be estimated by the students' major departments (using the proposed method) and aggregated by the department offering the course. Planning for elective courses is, by its nature, difficult and is not considered here. 5. SUMMARY This p a p e r proposes a methodology for deciding which courses and how m a n y sections of each to offer in an academic department. The procedure is based on the prerequisite and corequisite structure of the curricula, enrolment data and history of passing rates in each course. The m e t h o d is illustrated using an example problem. Acknowledgement--The authors would like to thank two anonymous referees for their constructive criticisms and valuable comments. REFERENCES 1. G. Andrew and R. Collins. Matching faculty to courses. College Univ. 46, 83-89 (1971). 2. P. Tillett. An operations research approach to the assignment of teachers to courses. Socio-Econ. Plann. Sci. 9, 101-104 (1975). 3. J. Breslaw. A linear programming solution to the faculty assignment problem. Socio-Econ. Plann. Sci. 10, 227-230 (1976). 4. R. McClure and C. Wells. A mathematical programming model for faculty course assignments. Decis. Sci. 15, 408-420 (1984). 5. R. McClure and C. Wells. Departmental planning in academia by decision support. College Univ. 61(2), 81-89 (1986). 6. S. Broder. Final examination scheduling. Commun. A C M 7(8t, 494-498 (May 1964). 7. ,D. Wood. A system for computing university examination timetables. Comput. J. 11(1), 41 (May 1968). 8. J. Grimes. 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