An annual problem for administration, be it at the high school, college or university level, is the assignment of teachers to courses. The general form of the ...
Socin-Econ. Ph.
Ei.. Vol. i0, pp. ‘2 _ 7, ‘30.
Pergamon Press 1976.
Prmted in Grear Britain
A LINEAR PROGRAMMING SOLUTION TO THE FACULTY ASSIGNMENT PROBLEM JON A. BRESLAW Department of Economics, Concordia University, Sir George WilliamsCampus, Montreal, Quebec, Canada (Received
1t F~b~~ry 1976; revised I4 July 1976)
Abstract-This paper investigates the problem of assigning faculty to courses at a university. A program is developed which is both efficient in that integer programming is not required, and effective, in that it facilitates interaction by administration
in determining the optimal solution. The results of some empirical tests are also reported.
I. I~RODU~ON An annual problem for administration, be it at the high
gorithm required an integer programming solution; this limits the efficacy of his model:
school, college or university level, is the assignment of teachers to courses. The general form of the problem is to find the solution to the following system: (a) The determination of the set of courses to be offered, and the number of sections in each. (b) The assignment of teachers to courses. (c) The assignment of courses and sections to rooms and time slots, and the allocation of teachers to sections. (d) The assignment of individual students to courses and sections, based on their requests, in such a way as to yield a minimum of conflicts. This system has to be solved simuItaneously to yield an optimal solution. A number of school scheduIing packages and techniques have been developed[l, 21. Indeed the solution to the Master Scheduling problem and the student allocation problem (parts c and d of the general Sorm of the problem) is available as a commercial service for schools and junior colleges; allocations satisfying 99% of student requests are regularty achieved even for large schools [3]. However, the allocation of courses to teachers, and the determination of how many sections of each course are given to a particular teacher (part b of the general problem) is taken as exogenous to the master scheduling problem. This allocation is usually done manually by the department chairman and/or the school principal, in consultation with the respective teachers. A number of models have been developed to efficiently solve this problem[2,4-6]. The state of the art is characterized by the work of Andrew and Collins[7]. An objective function, consisting of the weighted sum of teacher preference and instructor effectiveness (as evaluated by the departmental chairman) is optimized, subject to course assignment constraints (every course must be taught) and teaching load constraints (each instructor must teach a full work load). This is a standard transportation problem for which a number of efficient algorithms exist, and which, by the nature of the model, guarantees integer solutions. Tillett(61 expanded the model to include different preference weightings by ins~uctors to allow for the possibility that the preference of a teacher for a course could vary with the number of sections of that course assigned to that teacher. He also introduced a set of constraints relating to the maximum number of new preparations acceptable to each teacher. Tillett’s alSEPS VOL.
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“the major barrier to immediate practical application of the model is the fact that in all but very small departments, the computer time for the execution of the zero-one integer programming algorithm was prohibitively large”. This paper attempts to overcome this difficulty, and to develop a model that has particular applicability to universities. The particular scheduling needs of a university are considered in Section II, and a model is developed in Section III. Empirical testing is carried out in Section IV. IL UNIVERSPTY SCHEDULING PROBLEM Universities tend to operate somewhat differently from high schools, in that the master scheduling problem does not exist in the same way. Each department decides which courses it will offer, how many sections of each and when these sections will be offered. Effectively, this year’s program tends to resemble last year’s, both in the number and type of courses and the times of the sections. This fairly rigid structure ensures compatibility between different departments. At registration, each student decides which course and sections he wishes to take; since section times are already available, he can select a program which he knows is feasible, in that there are no time conflicts. Maximum section size is predetermined as a matter of policy, so that when a section is full, it is simply closed. If there is substantial excess demand for a particular course, an additional section might be opened, to be staffed by a part-time instructor; this procedure occurs relatively infrequently. The real problem is the allocation of faculty to courses prior to registration. Since there is a fair degree of change in faculty preference from year to year, as many instructors will want to teach new courses, rather than get stale, and given the normal staffing changes, visiting appointments, leave of absence, etc. the problem is not trivial. A number of considerations thus have to be taken into account when constructing a model for use in the university context. (1) The question of effectiveness is no longer valid in this situation-it is normally considered that if an instructor wishes to teach a course, then he will probably
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be as effective in that course as in any other course that he teaches. Presumably, in the extreme, a particular preference can be vetoed by the departmental chairman. (2) Faculty preference must be taken into account, and the preference weightings must be a function of the number of sections of a particular course that an instructor teaches. (3) Since the timetable is already determined, a number of constraints must be added to prevent possible conflicts. (4) The usual teaching load and section assignment constraints must be met. (5) One would like to apply such a package to a large department; this effectively prevents the use of zero-one integer programming because of the prohibitive cost. These are the purely technical considerations. However, past experience has shown that the application of such an O.R. package does not satisfy administration. After having made a trial run, one learns that the “optimal” solution is “not satisfactory”; it is politically unacceptable, socially unjust, or a number of other euphemisms for “it won’t do”. Some constraints, one learns, have been omitted; some bounds are not as rigid as they were first presented. This sort of experience is not new-Stimson and Thompson [8] clearly appreciated it: “Thus, at the end of it all, the operation researcher has the optimal solution to a problem in which the parameters, constraints and so forth are not optimally determined, and where his assumptions are not necessarily realistic in terms of the problems facing the administration.. . Hence the solution to a linear programming model.. should not be viewed as the solution to the problem. Rather it should be judged for its usefulness in a specific solution. The package technical problem does so in such involved in the administration in
described in Section III solves the of a university scheduling problem but a way that administration is actively problem, and hence effectively aids solving a real world problem.
IU. MODEL METHODOLOGY The academic assignment problem is a variant of the usual assignment problem, which itself falls within the set of problems categorized by the transportation problem. The transportation problem is profusely described in the literature[9]; the assignment problem is similarly well discussed[lO, 111. The problem consists basically of minimizing a costs matrix (or maximizing a benefits matrix) that results from any assignment choice, subject to the available staff and ensuring that all the jobs are filled. This problem could be solved using zero-one integer programming; however, as discussed above, this becomes infeasible for large problems. In fact, this is not necessary, since the ordinary SIMPLEX algorithm will give integer solutions to the transportation problem since the structural form of the constraint set guarantees that the extreme points of the feasible region will be integer. Consider the academic assignment problem. Let the variable Xii be the percentage of course i taught by faculty j. A zero value implies that faculty j does not teach course i at all, while a value of unity implies that the :The concept “course” will be taken to embody both the concept course and section-i.e. a particular time slot in which a particular study is taught.
course is totally taught by faculty j. Intermediate values are unacceptable, since they would employ that some of the course is taught by one instructor, and part by another-an inadmissable situation. Hence Xii must take the values of either zero or unity. pii is the preference rating for course i by faculty j. The university assignment problem can be formulated as a linear programming problem as follows: Max 2 C P+% subject to i= 1,2 ,...n
gxi=l i=l $Xij= i=L
6
(1)
j=1,2,...n
2X+1 iEt
j =1,2,...n k = 1,2,. . K
cxijsl ieg
j= 1,2 ,...m g=l,2,...G
xij30
i=l,2 j=l,2::::m.
(4) n
The objective function is the weighted sum of the X, for each course and faculty, where the weights are derived from the faculty preference ordering (for methodology, see below). The first constraint set (1) requires that each courset is taught once, i.e. that the total assignment value associated with each course must equal unity. Constraint set (2) requires that each instructor teaches his course load (Wj)-thus the total assignment value for the jth instructor is lI$ Since it is normally required that faculty split their work load equally between semesters, this constraint will apply twice for each instructor, where W, represents the respective semester work load. The classical assignment problem consists of these two constraint sets, together with non-negativity conditions. Constraint set (3) stipulates that an instructor may not teach more than one course in any one time slot (k); this constraint holds for all instructors and all time slots. The inequality exists since it is permitted for an instructor to teach no courses in a particular time slot. In practice, this constraint need only be considered for those time slots where there exists a possibility of more than one course being taught by a particular faculty member. Constraint set (4) permits an instructor to teach a maximum of one course from a particular cluster (g). Again the inequality is permitted, since no courses may, in fact, be selected from a particular cluster. Finally, constraint set (5) imposes non-negativity conditions on the Xii. Faculty list the courses they wish to teach in order of declining preference. If an instructor wishes to teach one section of a particular course (or indeed one course out of a number of courses), the set of sections (or courses) concerned is called a cluster. All the entities within a cluster naturally have equal preference weighting. Should an instructor wish to teach another section of the same course, but wishes this section to be assigned a lower preference value than the first section, then a cluster of desired sections is given lower down in the preference list. Half year courses received half weights.
A linear programming solution to the faculty assignment problem
The Simplex algorithm was used to solve the problem, and, as mentioned before, provides an integer solution. The addition of inequality type constraints (3) and (4) does not change this result (see Appendix). In this case, since we are dealing with a sparse matrix, the Revised Simplex algorithm was used. Thus a global optimum is guaranteed. This takes care of the technical side. The other requirement was an interactive capability for administration. This was achieved by programming a complete package, which carried out initial processing, linear programming and final processing, and which prints output at each stage. The initial processing produced a listing of timeslotsandfacultyteachingloads;alistingof preassigned courses by faculty (a preassigned course is one for which only one faculty has expressed a preference) for each semester; initial consistency checking; a listing of the remaining unassigned teaching load for each faculty by semester; a cross-listing of unassigned courses, their time slots and feasible instructors; and a timetable showing all courses and the respective instructor for the preassigned courses. Error messages indicated where no feasible solution was possible at any stage of the process. This initial processing, before a linear solution was attempted, not only allowed interaction between the O.R. staff and administration, but also saved a lot of computer time in attempting to solve problems for which no feasible solution existed. After a successful run, a listing of courses by faculty and semester, a full timetable and a normalized table of preference values for each faculty were produced. IV. EMPIRICAL
TESTING
The efficacy of a particular program in solving assignment problems is di~eult to measure, since each particular assignment problem differs from another, and so no standard exists. Two approaches were taken: a real world problem and a theoretical problem. Although neither of these approaches is ideal individually, the experience gained from solving these two problems, taken together, gives some idea as to the usefulness of this type of approach. A. A real world problem The department of economics at Concordia University (SGW), Montreal, has the usual timetable problem; manual solution, based on previous year’s experience, averages three months. The department is large enough to provide a reasonable test. Each faculty member was asked to give an ordered preference listing of the courses that he wanted to teach, of total length of twice his required teaching load-this provided sufficient slack, hopefully, for a feasible set to exist. The problem covered 85 courses, some of which were one semester, and the rest full year, 20 faculty, 20 time slots and 2 semesters. This appeared to be a relatively easy problem, at least in computer terms-an optimal solution was obtained in less than 6 seconds of CP time (CDC 6400). It was clear, however, that the most valuable output of the package were the listings produced in the initial processing prior to the linear programming solution. By giving administration information in a recognizable and useful form, that showed each stage of processing, a degree of communication developed that allowed far more interaction than is usually the case. This is borne out by the fact that by using these techniques, the timetable task was finished in two weeks-a distinct improvement.
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B. A theoretical problem The following theoretical problem was set up: forty courses are to be taught in the first semester, forty in the second and another forty are full year courses. Twentysix faculty each have a teaching load of 3 courses per semester, and an additional instructor has a load of 2 courses per semester. Thus total course load and total teaching load are equal in each semester. There are 20 time slots in each semester. To ensure a feasible solution a particular solution was chosen. Each course was then assigned a time slot randomly (based on a table of random numbers), with the exception that if a course in the particular solution was assigned a time slot which involved a conflict with a course that was also in the particular solution, then a new time slot was selected. Each faculty was assigned a total of nine courses-the particular solution, plus remaining courses, also chosen randomly. The order of the nine courses was also randomly determined; the order was presumed to signify the preference ordering. An optimal solution was obtained in 13 seconds (decimal) of CP time (201 iterations). The optimal solution differed from the particular solution, and obviously produced a higher valued objective function. For n = 120 (courses) and m = 54 (27 faculty, 2 semesters), this implies a permutation matrix of 6480 elements, which is by no means trivial. V. CONCLUSION
The results discussed here suggest that there are considerable benefits to be gained by using optimization methods in the solution of the university scheduling problem. It is perceived, however, that these gains do not accrue solely by utilizing an efficient algorithm, necessary though such an algorithm may be. Rather, these benefits come about to a large extent by the bringing in of administration during the O.R. process, rather than, as is the norm, of communicating before the process, and again afterwards. It appears that this interaction can only be achieved if the output is presented in such a way as to be precise, informative, and above all, familiar. O.R. techniques have appeared to be most effective in dealing with small specific micro models, where the system is well defined, and optimization takes place within this specific system. At the macro stage, where the system itself is not even a given, O.R. performance is dubious, to say the least. One can optimize a battlefield deployment but that hardly leads to the desired result if, in reality, one shouldn’t be fighting the war. if there is a qualitative nature to the world, be it the preservation of a species or the running of a school system, and if this qualitative nature is best appreciated by an administrator because perhaps of familiarity, or intuition, then it may well be that an interactive symbiosis of O.R. techniques coupled with administration “input” may yield better results than either one could do individually. Acknowledgements-The author gratefully acknowledges the encouragement and help he received from Morton Stelcner and Muriel Armstrong, both of Concordia University. REFERENCES I.
R. E. Holz, Computer-assisted scheduling. J. Ed. Data Processing 1, 36.41 (1964). 2. J. F. McNamara, Mathematical programming models in educational planning. Rev. Ed. Res. 41(S), 419-446 (1973).
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6.
1. 8.
9. 10.
11.
J. A. BRESLAW E. Harper (Programming consultant, Cogito, Inc.), Private communication (Nov. 1975). G. M. Andrew, Operations Analysis. University of Minnesota Press (1970). K. A. Fox and J. K. Sengupta, The specification of econometric models for planning educational systems. KY/&S 21, 665-693 (1968). P. I. Tillett, An operations research approach to the assignment of teachers to courses. Sot.-Econ. Plan. Sci. 9(3), 101-104 (1975). G. M. Andrew and R. Collins, Matching faculty to courses. College University 46, 83-89 (1971). D. H. Stimson and R. P. Thomoson, Linear proarammina. busing, and educational administration. Sot.-E~on.P[on. SA. S(4), 195-205 (1974). G. B. Dantzig, Linear Programming and Extensions. Princeton University Press, Princeton, N.J. (1963). P. S. Dwyer, Solution to the personnel classification problem with the method of optimal regions. Psychomefrika 19,l l-26 (1954). J. von Neumann, A certain zero-sum two person game equivalent to the optimal assignment problem. In Contributions to the Theory of Games II (Edited by H. W. Kuhn and A. W. Tucker), pp.5-12. Princeton University Press, Princeton, N.J. (1953). APPENDIX
In the usual linear programming problem, the linear constraints define a convex polyhedron in an n-dimensional space. The optimum point is found at a corner, or extreme point, of the polyhedron. The optimal value of the objective function defines a supporting hyperplane at this point. Even if we do not know the form of the objective function, we know that if the constraints are specified in such a way that the resulting polyhedron has integer values for all its extreme points, then the solution to the L.P. problem must also have an integer solution.
Consider a (one semester, for simplicity) problem, with n courses, m faculty, and r other constraints (inequalities). There are n X m variables (Xl,), n + m artificial variables and r slack variables. An initial feasible solution will eliminate all but one of the artificial variables from the basis, since the equality constraint set consists of only n + m - 1 independent equations-sum of course load equals sum of teaching load. The n + m + r equations form a basis, whose solution is composed of at the most of n + m t r non-zero components. If r of these components solve for the slack variables (i.e. all the slack variables belong to the basis), and one component solves for the remaining artificial variable, there will be left at the most n t m - 1 non zero components for the matrix X. Since the permutation matrix is m x n, there exists a row or column with only one non-zero element in it. Since the constraints are all integer, the element must also be integer. By removing that row (say), and changing the respective teaching load (W;), a new extreme point is reached involving n - 1 rows and at the most m columns. Repeating the argument, it is clear that, providing the slack variables enter the basis, that all the extreme points are integer. Initially, when the artificial variables all enter the basis, all the slack variables will have non-zero values, and hence also belong to the basis. Consider the case of the ith variable entering the basis, such that the kth slack variable changes from a non-zero value to zero (the inequality becomes an equality). However, when the ith variable enters the basis, some other variable, say the jth, must leave the basis, since the sum of the decision variables (i.e. non-slack variables) is constant; it equals the total course load. Hence the slack variables always remain in the basis, since no decision variable can be altered unless some other decision variable is also altered. Consequently, if the slack variables always remain in the basis, then, by the argument given above, the extreme points are guaranteed to be integer, and hence the optimal solution must also be integer.
