Course assignment problem with interval requested workload Phantipa Thipwiwatpotjana Department of Mathematics and Computer Science, Faculty of Science, Chulalongkorn University, Bangkok, Thailand, 10330 Email:
[email protected]
Abstract—While dealing with the course assignment of all instructors in the department of Mathematics and Computer Science, faculty of Science, Chulalongkorn University, we found out some flexibility in the requested teaching workload. Instructors had been asked to provide their requested teaching workload and preference subjects for each coming up semester. These data were used to create the course assignment by hand, which had no guarantee in the sense of optimality. More troubles happened when some instructors would like to change their promising teaching workload to be larger or smaller than their original requested teaching workload, due to various reasons. This creates an uncertain range of possible requested teaching workload and may lead to a different course assignment. The possible requested workload of each instructor has the sense of ignorance fuzzy/possibility to it. We apply some ideas in interval linear program to work with this research, which is an interval mixed binary linear program, to be able to inform the subjects and the amount of workload that each instructor would be assigned to teach under the objective of optimizing the overall preferences and extra/remaining workload using the pessimistic and optimistic approaches. Keywords: Course assignment problem; Interval linear program; Uncertainty.
I. I NTRODUCTION A course assignment problem is a decision making problem: maching subjects to instructors in order to verify an optimum under some objectives. The goals could be to minimize a total spend, to maximize the student learning process, etc. For this paper we have the objective to optimize the overall assignment preferences of the instructors in the department of Mathematics and Computer Science, faculty of Science, Chulalongkorn University. At the same time, we also try to minimize the extra and/or remaining amount of workload that instructors may receive. The department contains 58 instructors and the total subjects of 98 subjects categorized into two main disciplines: Mathematics and Computer Science. Some subjects may have multiple sections. We design the data sheet to be able to handle the teaching assignment of each section. Thus ‘a subject’ shown in this paper could mean a subject with only one section or a section of a subject if that subject contains more than one section. A total number of sections of 98 subjects is 120 sections. Each instructor requires to fulfill the total workload of 35 units. The workload of 7 units out of 35 is to serve on academic community services or any other service duties assigned by the department head. The rest is for the teaching
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and research workloads. The amount of teaching workload depends on each individual research production. Curtainly, the research outcome is a main criterion for evaluating each individual’s performance. However, in some situations, eg. personal/family illness and other accidents, some instructors may want to reduce the amount of their requested teaching workload after they know their assigned teaching workload. On the other hand, some instructors may face some situations causing them not to be able to finish their research as plan, which will effect their promising workload. Under these circumstances, the department still needs to be able to run smoothly. This leads to a question ‘what should be a reasonable assignment for each instructor, under some unknown/uncertain requested teaching workload?’. It should be a standard practice for any full-time instructors to be able to work during the working hours. Therefore, a course assignment problem should deal with the instructors’ preferences, without concerning about time slot at this moment. Moreover, an assignment done manually may cause bias. An integer linear program is another option to decrease this bias. A reasonable objective function that may reduce bias is to optimize the overall instructors’ preference. In addition, we also would like to reduce the over/under assigned workload that may occur. There are some articles concentrating on teaching assignment problem, eg. [6] and [7]. In [7], the authors applied the back-tracking with look-ahead forward checking method to their teaching assignment problem that has teacher preferences as soft constraints. Their solution is based on the fairness principle and the maximum sum of all satisfied preferences. However, this research was done by computer coding called the teaching assignment problem solver, which was developed to be able to access through the web site. A mathematical model provided for a teacher assignment problem in [6] had been solved using simulated annealing and tabu search in order to balance the teachers’ load. The other literature review relates to class-teacher timetabling problem. Many researchers applied graph coloring, heuristic/ meta-heuristic, case-based reasoning and fuzzy synthetic decision-making techniques to deal with the variety of timetabling problems, see [1], [2], [3], [4] and [8], for instances. In this work, as we concern about the unknown requested workload, we try to obtain assignment solutions using pes-
simistic and optimistic approaches with the help of some proved results in Section IV and a small modification our original model. We explain the information needed to formulate a reasonable binary linear program with interval requested workload for our course assignment problem, in the next section. The objective function and constraints are built in Section III, based on the given information to minimize the overall preference value and over/under assigned workload. Section IV concerns some useful theories for our interval course assignment problem. The model is run using CPLEX version 12.2 to find both optimistic and pessimistic solutions. The results and conclusion are in Sections V and VI, respectively. II. D ESCRIPTION FOR THE COURSE ASSIGNMENT PROBLEM In this section, we present the information and restrictions collected by the department which are used for planing instructors’ teaching courses. The instructors needed to provide their requested amount of workload and the ranks of preferred subjects. The rank of each instructor preference was verified by his/ her own specialist and/or research interest. The description of each instructor’s rank is provided in Table I. It could be more than one subject with the same rank. The fuzzy information on the preference will be our future study. Furthermore, we assume in this paper that all instructors have the same consistent judgement (without fuzzy) to clarify the rank of subjects they would like to teach. Each instructor may be asked to provide the weights representing his/her own ranking (in increasing order), within the range of 0 to 10,000. Therefore, the weights or values of rank could be evaluated as the average of the weights got from instructors, for simplicity. The average weights used in this work assume to be as presented in Table I. The requested teaching workload of each instructor includes Rank description A number one preferable subject to teach A preferable subject to teach An ok subject to teach A non preferable subject but ok to teach A non preferable subject A non preferable subject and does not want to teach
Rank 1
Value of rank 1
2 3 4
20 100 500
5 6
5,000 10,000
TABLE I T EACHING PREFERENCE RANK DESCRIPTIONS .
teaching regular classes, seminar and project/thesis advisor duties. The workload of each subject depends on its credit hours, enrollment numbers and class level. More details on this information, please see [9]. The standard constraints applied to the department teaching assignment system are as follows. 1) The number of teaching subjects for each instructor cannot exceed 3 subjects. 2) Only one teacher is required for one subject. 3) The department avoids to assign multiple teaching section of the same subject for an individual instructor.
The reason for this restriction is to reduce an unfairness among instructors. It is unfair if one person prepares only 1 subject but some others need to do more preparation. 4) Every instructor should meet his/her own requested amount of workload. However, the instructor’s requested workload could be presented as interval information due to some uncertainty. We will formulate the course assignment problem with interval workload using these constraints in the next section. III. F ORMULATION OF AN INTERVAL REQUESTED WORKLOAD COURSE ASSIGNMENT PROBLEM
In order to formulate a reasonable model of an interval requested workload course assignment problem, let us define variables and terms used in the model. th • Indices in the model: let i be the index of the i instructor, i = 1, 2, . . . , n and j be the index of the j th subject, j = 1, 2, . . . , m, where n = 58 and m = 120. th th • Teaching status of ththe i instructor to the jth subject: 1; the i instructor teaches the j subject xij = 0; otherwise. th th • Value of rank of the j subject with respect to the i instructor: cij . The value cij displays in Table I. th • The workload of the j subject: aj . The value of aj th depends on the j subject’s credit hours, enrollment numbers and class level. th • The requested workload of the i instructor: bi . The value of bi can be in the interval [18, 24.5]. th • Seminar, project and thesis duties of the i instructor: di . It is a known parameter workload, since instructors accepted to serve these duties before being assigned to teach. The ith instructor needs to fulfill the amount of teaching workload of bi − di . th • The excessive workload of the i instructor: δi ≥ 0. The excessive workload is the extra workload that the ith instructor has on top of the requested workload. th • The remaining workload of the i instructor: βi ≥ 0. The remaining workload is the remaining amount of the requested workload after deducting from the actual assigned workload. We can use these terms to formulate the standard restrictions as follows. 1) The number of teaching subjects each instructor cannot exceed 3 subjects: m xij ≤ 3, ∀i = 1, 2, . . . , n. j=1
2) Only one teacher is required for one subject. n xij = 1, ∀j = 1, 2, . . . , m. i=1
3) The department avoids to assign multiple teaching section of the same subject for an individual instructor. xij ≤ 1, ∀i = 1, 2, . . . , n j∈Jj
where Jj is the index set of multiple sections of the j th subject. 4) Every instructor should meet his/her own requested amount of workload. m aj xij − δi + βi = bi − di , ∀i = 1, 2, . . . , n. (1) j=1
Many circumstances may cause instructors to change their requested workload. For this reason, Constraint (1) could be adjust to be an interval constraint as follows, where [bi , bi ] refers to an interval of requested workload range [18, 24.5]. m
aj xij − δi + βi = [bi , bi ] − di , ∀i = 1, 2, . . . , n.
(2)
j=1
This interval restriction (2) means that the left-hand-side quantity could be equal to any value in the right-hand-side interval, in an actual circumstance. As our aim to satisfy the overall instructor preferences, a reasonable objective function could be minimize the overall preferences with high penalty of excessive and remaining workload. With this objective function, the model will try to assign teachers to subjects with lower ranks and reduce the over/under workload. Our objective function is stated as follows. m n n n min cij xij + M1 δi + M 2 βi , i=1 j=1
i=1
we know a particular value bpes ∈ [bi , bi ] that provides the i largest objective value (for a minimization problem) comparing with all other values in the interval. In a similar fashion, an optimistic solution is evaluated by using a value in the interval that provides the smallest objective value for a minimization problem. Before we continue with pessimistic and optimistic approaches, let us provide the theorems used to evaluate these approaches. We relax the binary variables in (3) by given a general problem with interval requirements and nonnegative decision variables as follows. min t T x
(4) s.t. A1 x ≥ p, p , A2 x ≤ s, A3 x = r, x ≥ 0. The interval constraint in (4) also could be an ‘=’
constraint or ‘≤’ constraint, as well. For a given p ∈ p, p , let us
define Ωp = x | A1 x ≥ p, p , A
2 x ≤ s, A3 x = r, x ≥ 0 . Suppose that Ωp = ∅, ∀ p ∈ p, p . Definition 4.1: An optimistic solution to Problem (4) is an optimal solution to the problem min tT x, xmin ∈Ω
p ∈ p, p
while a pessimistic solution to Problem (4) is an optimal solution to the problem tT x. max xmin ∈Ω
i=1
where M1 and M2 are penalty terms of excessive and leftover workload, respectively. Having positive remaining workload is more severe since it implies that an instructor could not perform his/her duty as promising. Thus, M2 should have more penalized weight than M1 . It would be the best if this model provides an optimal solution with value zeros of these workloads.
p ∈ p, p
min t T x
s.t. A1 x = p, p , A2 x ≤ s, A3 x = r, x ≥ 0, can be found by solving the linear program min t T x s.t. A1 x ≥ p, A1 x ≤ p, A2 x ≤ s, A3 x = r, x ≥ 0.
ASSIGNMENT PROBLEM
j = 1, 2, . . . , m.
By changing any [bi , bi ] to be just an element bi in the interval, Model (3) becomes a mixed binary linear programming problem. Model (3) is an interval mixed binary linear programming problem. With the unknown requested workload, reasonable solutions to the problem could be interpreted using pessimistic and optimistic approaches. A pessimistic solution is found if
p
Theorem 4.1: (see Chapter 3 in [5]) An optimistic solution of the problem
IV. I NTERVAL LINEAR PROGRAM WITH COURSE Our mathematical model of an interval requested workload course assignment problem described in the previous section is stated as Model (3). n m n n ⎫ min δi + M2 βi ij xij + M1 i=1 j=1 i=1 i=1 ⎪ cm ⎪ ⎪ s.t. xij ≤ 3, i = 1, 2, . . . , n ⎪ j=1 ⎪ n ⎪ ⎪ xij = 1, j = 1, 2, . . . , m ⎬ i=1 xij ≤ 1, i = 1, 2, . . . , n (3) j∈Jj m ⎪ aj xij − δi + βi = [bi , bi ] − di , i = 1, 2, . . . , n⎪ ⎪ j=1 ⎪ ⎪ δi , βi ≥ 0, i = 1, 2, . . . , n ⎪ ⎪ ⎭ xij ∈ {0, 1} , i = 1, 2, . . . , n
p
Proof : See the proof in [5]. Theorem 4.2: Suppose that Ωp = ∅, ∀ p ∈ p, p and min tT x is bounded for each p. Then, an optimistic solution
x ∈ Ωp
of Problem (4) is an optimal solution to the problem min tT x min tT x ≡ s.t. A1 x ≥ p, A2 x ≤ s, A3 x = r, x ≥ 0. x ∈ Ωp Similarly, a pessimistic solution of Problem (4) is an optimal solution to the problem min tT x min tT x ≡ s.t. A1 x ≥ p, A2 x ≤ s, A3 x = r, x ≥ 0. x ∈ Ωp
Proof : Let p ∈
p, p . The problem min tT x x ∈ Ωp
has the
following corresponding dual problem: max s.t.
⎫ yT p + uT s + wT r ⎬ AT1 y + AT2 u + AT3 w ≤ t ⎭ y ≥ 0, u ≤ 0, w is free.
(5)
Since Ωp = ∅ and min tT x is bounded for each p ∈ p, p , x ∈ Ωp
its dual problem is also feasible and bounded. Let (yp , up , w
p) be an optimal solution for Problem (5), for each p ∈ p, p . ≤ ≤ ≤ ≤
yTp yTp yTp yTp yTp
p + uTp s + wTp r p + uTp s + wTp r, p + uTp s + wTp r, p + uTp s + wTp r, p + uTp s + wTp r,
(y ≥ 0 and p ≤ p) (dual optimality) (y ≥ 0 and p ≥ p) (dual optimality).
By the strong duality theorem, we can conclude that an optimistic solution to Problem (4) is an optimal solution to the problem min tT x and a pessimistic solution to Problem x ∈ Ωp
(4) is an optimal solution to the problem min tT x. x ∈ Ωp
With the ‘≤’ inequality adjustment of Constraint (2), Model (3) becomes Problem (6). We can apply Theorem 4.2 to Problem (6) as explained in Corollary 4.1, even though our problem contains binary variables (teaching status). Corollary 4.1: Optimistic and pessimistic solutions to the interval linear problem (6) n m n n ⎫ min δi + M2 βi ij xij + M1 i=1 j=1 i=1 i=1 ⎪ cm ⎪ ⎪ s.t. xij ≤ 3, i = 1, 2, . . . , n ⎪ j=1 ⎪ n ⎪ ⎪ xij = 1, j = 1, 2, . . . , m ⎬ i=1 xij ≤ 1, i = 1, 2, . . . , n (6) j∈Jj m ⎪ aj xij − δi + βi ≥ [bi , bi ] − di , i = 1, 2, . . . , n⎪ ⎪ j=1 ⎪ ⎪ δi , βi ≥ 0, i = 1, 2, . . . , n ⎪ ⎪ ⎭ xij ∈ {0, 1} , i = 1, 2, . . . , n j = 1, 2, . . . , m
can be found by solving two linear programs with bi and bi for all i, respectively. Proof : Let us relax Problem (6) to be a linear program by choosing some specific xij to be equal to 1 and the others as zeros. The related problem is presented below in the system (7). Let Λ be the set of all indices ij, where i ∈ {1, 2, . . . , n} and j ∈ {1, 2, . . . , m}. Moreover, let Λ1 , Λ2 ⊆ Λ such that Λ1 ∩ Λ2 = ∅ and Λ1 ∪ Λ2 = Λ. n m n n ⎫ min δi + M2 βi ⎪ ij xij + M1 i=1 j=1 i=1 i=1 cm ⎪ ⎪ ⎪ s.t. xij ≤ 3, i = 1, 2, . . . , n ⎪ j=1 ⎪ n ⎪ xij = 1, j = 1, 2, . . . , m ⎪ ⎪ i=1 ⎬ xij ≤ 1, i = 1, 2, . . . , n j∈Jj m (7) aj xij − δi + βi ≥ [bi , bi ] − di , i = 1, 2, . . . , n⎪ ⎪ j=1 ⎪ ⎪ δi , βi ≥ 0, i = 1, 2, . . . , n ⎪ ⎪ xij = 1, ij ∈ Λ1 ⎪ ⎪ ⎪ xij = 0, ij ∈ Λ2 ⎭ xij
≥
0, ij ∈ Λ.
Even though Problem (7) has a lot of redundant constraints, we still can apply Theorem 4.2 to it. This implies that for any sets Λ1 , Λ2 ∈ Λ, we could use b and b to evaluate the optimistic and pessimistic objective values of (7). One particular pair of the sets Λ1 , Λ2 ∈ Λ provides an optimisitic solution to (6), hence, we use b instead of the interval [b, b] in (6) to evaluate this particular pair of Λ1 , Λ2 . A pessimistic solution could be reached in a similar fashion. An optimistic solution of (3) also can be found using the modification of Theorem 4.1 with binary restriction in a similar pattern as in the proof of Corollary 4.1. However, it is
not easy to evaluate a pessimistic solution of (3), in general. For our assignment problem, we try to use Theorem 4.2 to find a pessimistic solution of (3), by looking into the details if we know a pessimistic solution (xpes , δ pes , βpes ) to Problem (6) with the greater than or equal to inequality interval constraint. Please note that the δ and β of the model (6) may not be representing the real over/under workload as ones in the model (3). The solution (xpes , δ pes , βpes ) of the modified model (6) results the following two cases. pes pes Case 1: m and βipes are j=1 aj xij > bi − di , ∃i then δi both zero. However, the real overload with respect to xpes will not be zero and xpes may not be a pessimistic solution for Problem (3), in general. m Case 2: j=1 aj xpes ≤ bi − di , ∀i then βi = bi − di − ij m j=1 aj xij and δi = 0. We can change the inequality ‘≥’ interval constraint to ‘=’ as in (3), and the pessimistic objective value and solution of Problem (3) are still the same as ones for the pessimistic of the modified model (6). We are fortunate enough that our data, by running the modified model (6) using CPLEX version 12.2, results in the Case 2 above. Hence, we obtain a pessimistic solution to our problem. V. O PTIMISTIC AND PESSIMISTIC SOLUTIONS TO THE INTERVAL REQUESTED WORKLOAD COURSE ASSIGNMENT PROBLEM
We simplifies the number of subjects of each preference under the pessimistic and optimistic approaches in Table II by using the penalties M1 = 5 × 104 and M2 = 6 × 104 . Rank 1 2 3 4 5 6
Number of subjects Optimistic Pessimistic 66 66 17 17 9 9 5 5 2 2 21 21
TABLE II O PTIMISTIC AND PESSIMISTIC NUMBERS OF SUBJECTS IN EACH RANKING PREFERENCE USING THE PENALTIES M1 = 5 × 104 AND M2 = 6 × 104 .
Rank 1 2 3 4 5 6
Number of subjects Optimistic Pessimistic 69 66 18 17 11 9 6 5 1 2 15 21
TABLE III O PTIMISTIC AND PESSIMISTIC NUMBERS OF SUBJECTS IN EACH RANKING PREFERENCE USING THE PENALTIES M1 = 6, 000 AND M2 = 6 × 104 .
Please note that since M1 is greater than the highest value of rank cij , we would expect the small δ terms. It turns out that both solutions are the same, when using the penalties
M1 = 5 × 104 and M2 = 6 × 104 in our model. However, we do not suppose to assign the instructors according to Table II, since there are too many subjects with the rank 6. To be more reasonable (according to the rank description in Table I), M1 should be less than 104 but should not be smaller than value of the rank ‘5’. In this case, it means that some instructors may need to teach more than their requests in order to maintain the teaching quality. Let us apply M1 = 6, 000. The result is shown in Table III. In addition, we summarize the pessimistic and optimistic working loads for each instructor of Problem (3) with the penalties M1 = 6, 000 and M2 = 6×104 in Table IV. Table IV tells us that if our priorities are to minimize the overall preferences, reduce the overload, but cannot tolerate the remaining workload, it is sufficient and significant to conclude that the department does not need to spend time asking for the requested workload. No matter how much the workload the instructors would like to work, their actual workload will not be improved that much. Moreover, since our aims do not focus on the total of the least preference subjects, we should not choose M1 much lower, e.g. between the values of ranks ‘4’ and ‘5’, eventhough it results in the small number of the last rank subjects. i 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
Lo 28.50 23.77 23.25 22.27 24.70 22.00 24.50 23.75 24.50 24.20 24.50 22.02 24.00 23.03 24.20 24.12 22.50 24.50 24.00 24.50
Lp 28.50 23.77 23.25 22.27 22.70 24.00 24.50 23.75 24.50 24.20 24.50 22.02 24.00 23.03 24.20 24.12 24.00 24.50 24.00 24.50
i 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40
Lo 24.20 22.97 24.00 24.00 24.00 24.00 23.10 24.00 24.00 25.03 24.00 25.07 24.03 24.10 23.00 23.50 23.97 24.50 23.50 24.00
Lp 24.20 22.97 24.00 24.00 24.00 24.00 23.10 24.00 24.00 23.03 24.00 23.57 22.53 24.10 23.00 23.50 24.47 24.50 22.00 24.00
i 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58
Lo 24.75 24.00 24.50 22.50 22.00 22.75 18.50 24.52 23.57 21.58 21.75 21.85 22.87 24.27 22.50 21.80 24.00 24.00
Lp 23.25 24.00 24.50 22.50 22.00 22.75 24.50 24.27 23.57 21.58 21.75 22.60 24.37 24.27 22.50 21.80 24.00 24.00
TABLE IV P ESSIMISTIC AND OPTIMISTIC WORKLOAD OF EACH INSTRUCTOR : i := THE iTH INSTRUCTOR , Lo := OPTIMISTIC WORKLOAD AND Lp := PESSIMISTIC WORKLOAD .
VI. C ONCLUSION This research investigates the importance of having the uncertainty in the requested workload of the teaching assignment problem when the aims are to minimize the overall preferences and reduce over/remaining workload. We apply the known and modified theorems (Theorems 4.1 and 4.2) to the problem in order to find the optimistic and pessimistic solutions of the problem. The result shows that both solutions are not significantly different. Hence, the department may save
time by droping the uncertainty option and use one of these solutions to assign the subjects for the instructors. ACKNOWLEDGMENT The author would like to thank Miss Saraprang and Miss Kiang-ia for helping with the CPLEX program. This work is under the Interdisciplinary Center for Computational Sciences Chulalongkorn university (ICCC). R EFERENCES [1] E. R. Burke, B. MacCathy, S. Petrovic and R. Qu, ‘Multiple-retrieval case-based reasoning for course timetabling problems’, Journal of the Opeartional Research Society, 1-15, 2005. [2] E. R. Burke, B. McCollum, A. Meisels, S. Petrovic and R. Qu, ‘A graphbased hyperheuristic for educational timetabling problem’, European Journal of Operational Research, Vol. 176(1): 177-192, 2007. [3] S. Daskalaki and T. Birbas, ‘Efficient solutions for university timetabling problem through integer programming’, European Journal of Operational Research, Vol 160(1): 106-120, 2005. [4] Y. Duan, Y. Zhong and Y. Li, ‘Application research on FSDM-based GA in optimizing curriculum schedule model in universities’, Fuzzy Information and Engineering, No. 2: 217-228, 2012. [5] M. Fiedler, J. Nedoma, J. Ramik, J. Rohn and K. Zimmermann, ‘Linear optimization problems with inexact data’, Sringer, New York, 2006. [6] A. Gunawan and K. M. Ng ‘Solving the teacher assignment problem by two metaheuristics’, International Journal of Information and Management Sciences, No. 22: 73-86, 2011. [7] A. Hmer and M. Mouhoub, ‘Teaching assignment problem solver’, N. Garcia-Pedrajas et al. (Eds.): IEA/AIE 2010, Part II, LNAI 6097, Springer, Berlin Heidelberg: 298-307, 2010. [8] S. R. Hu, Y. Deng and Z. Wang, ‘Graph theory research based on college time-table system’, Computer Engineering and Applications, Vol 10(4): 221-223, 2002. [9] B. Sa-ngimnet and J. Eabsrangky ‘The optimum teaching schedule to satisfy instructor requirements’, Senior Project, Department of Mathematics and Computer Science, Faculty of Science, Chulalongkorn University, 2012.
