Mathematical and Computer Modelling 52 (2010) 930–941
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Exam scheduling: Mathematical modeling and parameter estimation with the Analytic Network Process approachI Mujgan Sagir a,∗ , Zehra Kamisli Ozturk b a
Industrial Engineering Department, Eskişehir Osmangazi University, Eskişehir, Turkey
b
Open Education Faculty, Anadolu University, Eskişehir, 26470, Turkey
article
info
Article history: Received 25 May 2010 Accepted 25 May 2010 Keywords: Invigilator-exam assignment Analytic Network Process Mathematical model
abstract An invigilator is a person who supervises students during examinations in educational systems. Assigning invigilators to exams is an important phase in the exam scheduling process. The problem has its own constraints and a multi-objective structure. Hardly any mathematical models have been used in the past. Mostly, heuristic approaches are used to assign invigilators to exams. The Analytic Network Process (ANP) with dependence and feedback is a general framework for a detailed analysis of societal, governmental and corporate decisions that is available today to the decision maker. Within the ANP networks of influence one includes all the factors and criteria, tangible and intangible, which have bearing on making a best decision. In this paper, an ANP model is used to prioritize the objectives of the invigilator-exam assignment problem. The model has ten clusters. All the participants and the criteria with their interrelations are defined. Different objectives of the problem are treated as alternatives in this model. Relative weights are obtained for the elements in the ANP model by using the Super Decisions software. Then we solve a previously developed nonlinear optimization model with several constraints using these weights. © 2010 Elsevier Ltd. All rights reserved.
1. Introduction Examination scheduling is a highly complex combinatorial problem consisting of NP-complete subproblems [1]. Assigning invigilators who may be faculty or graduate students to exams in educational institutions subject to constraints is one aspect of this problem. But it is often done separately, prior to, or after, scheduling examinations to time slots and to rooms [2–5]. This study deals with some aspects of the invigilator-exam assignment as part of an ongoing research effort related to the educational timetabling problem. Conventionally, the invigilator assignment problem used to be solved manually at the Department of Management, Eskişehir Osmangazi University, where the scheduler assigned the exams to invigilators taking into account the invigilators’ available time slots. Assigning as equal as possible a number of exams to each invigilators was the only criterion to satisfy in this manual process. But exam durations vary from 30 to 120 min and some exams are given on weekends. These and other variations affected the invigilators’ preferences and performance and were not considered systematically. To improve the assignment process, a mathematical model was developed, in which the exams were weighted to satisfy three objectives. The model considered the number of exams assigned to each invigilator, the duration of the exams and the I This study is supported by Eskişehir Osmangazi University scientific research projects committee (ogubap-project no: 200815009) and by the Scientific and Technological Research Council (Tubitak) of Turkey. ∗ Corresponding author. Tel.: +90 2222393750; fax: +90 2222391775. E-mail addresses:
[email protected],
[email protected] (M. Sagir),
[email protected] (Z.K. Ozturk).
0895-7177/$ – see front matter © 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.mcm.2010.05.029
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day of the week on which they fell. Formerly, through discussions in the department, arbitrary numbers such as 2, 3, and 4 were assigned to the exams as weights. The more burden an exam places on an invigilator, the higher the number assigned to it. These numbers, though arbitrary, were considered to be ratio scale numbers with 2 twice as difficult as 1, 4 twice as difficult as 2 and so on. Using a mathematical modeling optimization framework an optimum solution was obtained and, compared to the manual solution, was found to satisfy the invigilators better and is described in this paper [6]. This new method of assigning invigilators, however, still had some drawbacks. In this paper we suggest an improvement. Midterm and final exams had already been scheduled by the Department of Management. First we assigned the invigilators using arbitrarily assigned ratio scale numbers for the exam weights as we had done before. Second we derived new weights for the exams using an Analytic Network Process (ANP) model that included such factors as duration, day, type and other things. Using the new more rationally derived weights for the exams we solved the optimization model again. The two sets of invigilator assignments are then compared. The Analytic Network Process used to derive the weights is described below. With the ANP model we were able to include both inner and outer dependencies among the factors. To explain the eventual contribution of this study we use the numerical outcomes of the two models and then determine which one is better all around. One of the objectives was to minimize the maximum (the worst) individual total load r1 (the total weight of the exams invigilated by an individual). Its value is 0.44 for the first model and 0.45 for the second model. For different values of the parameters (that is, the exam weights) one obtains different optimum values for the objective function. However, if one value is less than another value for a different set of parameters, it does not mean that the solution is more realistic and satisfactory to those who have to live with it. This shows that the first set of exams and objective weights could be misleading, because individual loads may actually be better in the second set (or let us say that the worst one in the first set is actually even worse than it is in the second set). This is a noteworthy observation that one needs to appreciate. Fudging the optimization constraints of an optimization problem does not guarantee the realism of the solution simply because its objective function has a smaller value. We are able to calculate them in a more realistic way by using the ANP model. The paper is organized as follows: Section 2 introduces the first mathematical optimization model of the problem together with its solution. The Analytic Network Process (ANP) approach is briefly summarized. The ANP model developed to obtain the exam and objective weights is discussed in Section 3. Section 4 provides a comparison of the two models and an explanation as to why the assignments obtained by the second model have produced greater satisfaction than both the first model and the earlier manual solution.
2. Mathematical model of the problem—the solution without ANP weights A multi-objective mixed integer mathematical model has been developed for an invigilator-exam assignment problem with the following model parameters, decision variables, constraints and objectives [6]. Sets and parameters: I = { 1, . . . , m} Invigilators J = { 1, . . . , n} Examinations T = { 1, . . . , k} Timeslots U = {(i, t )I ∃ i ∈ I , ∃t ∈ T } The set of unavailable time slots for invigilators gj : The number of invigilators required for the jth exam aj : Weight of the jth exam sjt : 1 if exam j is scheduled during time slot t, 0 otherwise mit : 1 if invigilator i is not available at time t pij : 1 if invigilator i is pre-assigned to course j; 0 otherwise cit : Assignment cost of ith invigilator to tth time slot wi : Load ratio of the ith invigilator Decision variable:
yij = min
1, 0,
if the ith invigilator is assigned to jth exam otherwise
XXX i
t
cit yij sjt + r1 + r2
j
subject to
X j
yij sjt ≤ 1,
∀ (i, t )
(C1)
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X
yij = gj ,
∀j ∈ j
(C2)
j
yij = pij ,
X
∃(i, j)
(C3)
yij sjt = mit − 1,
∃(i, t ) ∈ U
(C4)
j
X
yij aj ≤ wi r1 ,
∀ (i)
(C5)
∀ i, ∃t .
(C6)
j
X
yij sjt ≤ r2 ,
j
(C1) presents the clash-free requirement (no invigilator can be assigned to two exams at the same time). (C2) guarantees that the required number of invigilators is assigned to each exam. Pre-assignments may be needed in which specific invigilators are assigned to the exams of specific courses and these are performed via (C3). An invigilator is allowed to have restrictions on specific time slots which are then considered to be disabled for him. For this purpose, in (C4), mit is defined as the parameter that takes the value 1 when invigilator i is not available during time t, in other words when (i, t ) ∈ U. To prevent the assignment procedure from doing this, 1 is subtracted from mit to guarantee that there will not be such an assignment. The reader may think that it would be enough to simply construct (C4) as:
X
yij sjt = 0,
∃(i, t ) ∈ U .
j
But because the time period preferences of invigilators are gathered via web based user interfaces, it is essential to store the related preference data in a parameter called mit for this case. Then the system uses the data that were entered to construct the mathematical model. It is assumed that U 6∈ ∅. The other preference parameters, cit , are used directly as they are entered into the web based system. As the preference value of being assigned to exam i at time slot t , cit gets either the value 1 or 2. On the other hand, any unavailable time slot t for an invigilator i is indicated by assigning 1 to mit . This represents an impossible assignment because of the unavailability of the related invigilator. In addition the number of duties should be fair by taking into account the number of times an individual is appointed as an invigilator (C5). Besides, the number of duties at unavailable time slots should be minimized. (C5) and (C6) are the linearization of: P r1 = min{maxi ( j yij aj )} which minimizes the maximum individual load over the feasible solution set, and r2 = min{maxi ( j yij sjt )} ∃(i, t ) which minimizes the maximum individual total assignment at undesired time slots. The positive decision variables r1 and r2 are used to linearize the min-max structure in the model. Thus the general variant of the exam scheduling problem can be formulated to obtain solutions when the following conditions are satisfied We would like to remind the reader that an objective function that is a weighted combination of several different objectives of different kinds has an optimum overall solution from which usually one cannot hope to extract the optimum solution for each of the objectives were it to be considered alone with the same constraints. That optimum would be in a sense a better one. But that is the nature of a problem with combined objectives. A feasible solution is one which completely satisfies all the hard constraints. The model has a multi-objective structure. The objectives are to minimize the assignment cost, the total maximum load of invigilators and maximum total assignment at undesired time slots, simultaneously. The weighted sum method is used to scalarize them; though with this kind of sum, the approach does not guarantee finding all pareto-optimal solutions due to the fact that the necessary convexity conditions do not hold. And this is because of the 0–1 decision variables that the problem has. Fortunately, there are studies in the literature that solve the problem with this kind of scalarization [7]. We have tested this proposed model using the invigilator assignments for final and midterm exams at Eskişehir Osmangazi University’s Industrial Engineering Department. There are four time slots reserved for examinations each day. The invigilators are teaching assistants, mostly graduate students. They are categorized as either experienced or new. A higher priority, which is translated into less duty, is assigned to the experienced invigilators to give them more time for their research activities. The optimal solution is obtained rapidly by using GAMS with CPLEX solver. The optimal solution with the proposed invigilator assignments, are summarized and compared with the current invigilator assignments for the Spring_midterm exams and Spring_final exams. For the Spring_midterm exams, for both the proposed and current assignments, the new invigilators must each monitor from 12 to 14 exams while the experienced ones monitor 7. With the proposed solution the total load for each individual is more equal (it changes from 32 to 36 in the proposed solution while it changes from 32 to 38 in the current solution).
P
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For Spring_final exams, the total number of examinations for invigilators changes from 10 to 12 for the current outcome while it is obtained as either 10 or 11 for the new ones, and 5 for the experienced invigilators for the proposed assignment. The invigilators are equally loaded (29 and 14 for the new and experienced ones) in the proposed solution also for this exam term. The current loads for this term are obtained from 27 to 31 for the new ones while these values are 11 and 16 for the experienced ones for this term. Here, the current assignment was done manually and the proposed one was done by the first mathematical model which used arbitrarily assigned numbers as course weights. It is seen that with the proposed solutions, dissatisfactions by invigilators decline. Actually, proposed schedules for both exam terms present perfectly loaded schedules under the current parameter set [6]. Though we have an optimal solution, different objectives exams for this mathematical model have not been weighted in a systematic way previously. For exam weights (aj ) 2, 3 and 4 are used and the objectives have been considered as equally weighed. The next section briefly introduces the ANP methodology. An ANP model to prioritize the model objectives is introduced next. We should mention that the mathematical model has the parameter aj as the weight of the jth exam which was just an arbitrary number (2, 3 or 4) in the first mathematical model, and then we calculate this weight aj by using ANP outcome in the following section. 3. The Analytic Network Process model for estimating exam and objective weights A robust measurement theory that applies paired comparisons to tangibles and intangibles within a hierarchic structure is the Analytic Hierarchy Process (AHP). The AHP depends on measurement values and on judgments elicited from individuals and from groups, in case many people are involved in the decision. Decision makers who use the AHP first structure the problem into a hierarchy: the top level is the overall objective and the bottom level includes the action alternatives or the alternative plans that would contribute positively or negatively to the main objective through their impact on the criteria in the intermediate levels of the hierarchy [8,9]. Priorities are derived from pairwise comparison judgment matrices in the form of the normalized principal eigenvector of each matrix of judgments. These vectors are used in a systematic weighting operation from the top to the bottom level to synthesize the weights of the alternative outcomes. The AHP is conceptually easy to use; however, its strict hierarchical structure cannot handle the complexities of many real world problems. The AHP assumes independence of the elements in each level and also of those in a higher level from lower level elements. To deal with the dependence of such elements within a level or between levels, T. Saaty developed, in addition to the AHP, the Analytic Network Process (ANP), of which the AHP is a special case. The ANP represents a decision making problem as a network of criteria and alternatives, grouped into clusters. All the elements in the network can be related in any possible way, i.e. a network can incorporate feedback and interdependence relationships within and between clusters. This provides a means to more accurately model complex decisions. The influence of the elements in the network on other elements in that network can be represented in a supermatrix. This new concept consists of a two-dimensional cluster-by-cluster and element-by-element matrix in which the eigenvectors from the judgment matrices are arranged [10]. In the ANP not only does the importance of the criteria determine the importance of the alternatives, as in a hierarchy, but also the alternatives themselves are used to determine the importance of the criteria. Two bridges, both strong, but the stronger is also uglier, would lead one to choose the strong but ugly one unless the criteria themselves are evaluated in terms of the bridges, and strength receives a smaller value and appearance a larger value because actually both bridges are strong enough. Feedback enables us to factor the future into the present to determine what we have to do to attain a desired future. In Fig. 1, we exhibit a hierarchy and a network. As we said before, a hierarchy is comprised of a goal and of levels of elements and connections between these elements [11]. The connections are oriented only to elements in lower levels. A network has clusters of elements, with the elements in one cluster connected to elements in another cluster (outer dependence) or within the same cluster (inner dependence). A hierarchy is a special case of a network with connections going only in one direction. There are two kinds of influence: outer and inner. In the first, one compares the influence of elements in a cluster on elements in another cluster with respect to a control criterion (a criterion such as economic, political, social influences, in terms of which the comparisons are made). In inner influence one compares the influence of elements in a group on each other. For example if one takes a family of father, mother and child, and then considers them one at a time. For the child one asks who contributes more to the child’s survival, its father or its mother, itself or its father, itself or its mother. In this case the child is not so important in contributing to its survival as its parents are. But if we take the mother and ask the same question as to who contributes to her survival more, herself or her husband, herself would be higher, or herself and the child, again herself would be the most important. Another example of inner dependence is in the making of electricity. To make electricity the electric industry needs steel to make turbines, and also needs fuel. We have three industries to consider: the electric industry, the steel industry and the fuel industry. When asked, ‘‘What does the electric industry depend on more to make electricity, itself or the steel industry?’’. The answer is that steel is more important. When asked if it depends more on itself or the fuel industry, the answer is that the fuel industry is much more important. When asked if it depends more on the steel or the fuel industry, fuel is more important. The electric industry does not need its own electricity to make electricity. It needs fuel. Its electricity is only used to light the rooms, which it may not even need. A supermatrix along with an example of one of its general entry matrices are shown in Fig. 2. For example, the component (C1) in the supermatrix includes all the priority vectors derived for nodes that are ‘‘parent’’ nodes of the (C1) cluster, which means the elements in (C1) influence some or all the elements that feed into (C1).
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Fig. 1. How a hierarchy compares to a network.
Fig. 2. The supermatrix of a network and detail of a component in it.
In the ANP we look for steady state priorities from a limit supermatrix. To obtain the limit we must raise the matrix to powers. The limit may not converge unless the matrix is column stochastic that is each of its columns sums to one. To ensure stochasticity of the matrix, one needs to compare the influence of all the clusters on each cluster with respect to the control criterion that underlies the comparisons from which the priorities in the supermatrix are derived. Powers of the matrix capture all transitivities of an order that is equal to the corresponding power of the supermatrix. All order transitivities are captured by the sequence of powers of the matrix. The limit of the priorities obtained by summing and normalizing the rows of each power and then taking the average of the resulting vectors, according to Cesaro Summability, is simply equal to the priorities derived from the limit of the powers of the supermatrix. The outcome of the ANP is nonlinear and complex. In the next section introduces the ANP model to prioritize the problem’s objectives [12]. A single level but detailed network model is developed to prioritize the objectives of the scheduling problem. The objectives are listed in alternatives cluster. The three objectives explained in the previous section are considered here: (1) minimize the maximum (the worst) individual total load of invigilators, (2) minimize the maximum (the worst) individual total number of assignment of invigilators to undesired time slots such as weekends and (3) minimize the total assignment cost. Clusters of elements that affect these objectives are constructed according to influence groups and issues related to them. Ten clusters are considered including the alternatives. They are: Students, faculty, invigilators, administration, courses, exam duration, exam day, exam time slots, exam type and alternatives. The elements in the clusters related to the exams are defined as 60, 90 and 120 min, for the exam duration from Monday to Saturday, 9 am–11 am, 11 am–1 pm, 2 pm–4 pm, and 4 pm–6 pm are considered for the exam time slots, and in addition these criteria are also considered: open book, without answer sheet, paper based, and test based for the exam type. Next, the connections among elements and clusters are determined by the decision maker. Fig. 3 represents the ANP model structure with a special screen view from the software, Super Decisions when the mode to examine the connections is selected for min-max assignment to undesired time slots criterion in the Alternatives cluster. The elements written in bold in other clusters are those defined as the influencers of the selected criterion. It is seen that min-max assignment to the undesired
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Fig. 3. Analytic Network Process model.
time slots criterion is influenced by all the elements in the exam time, exam day, and invigilator clusters. Besides this faculty requests for related exams in the Faculty cluster and providing required resources for the assignment in the Administration cluster are for other factors that influence the selected criterion. Similarly, as each criterion is selected, the software provides the decision maker with an opportunity to see the connections of the elements that must be compared with respect to that criterion. In the next step, paired comparisons are performed to obtain the relative priorities of the elements and also of the clusters. Which factor influences the objective of min-max assignments to undesired time slots more: Faculty requests on related exams or being an experienced invigilator? Since faculty members prefer their teaching assistants to invigilate their own courses, – as a request on related exam – this parameter is considered as more effective when making comparisons with respect to the experience level of an invigilator. Clusters that have any links to their factors from the factors in a given cluster must be compared for their impact on the given cluster. As an example Fig. 4 gives a screen view of the paired comparisons of clusters that must be compared for their impact on the Course cluster. According to the decision maker, the Faculty cluster has a twice greater effect on the Course cluster than that does the Administration cluster. This is because, relatively speaking, the faculty has more rights than the administration on describing the course and course exams. On the other hand, when compared with students, the administration has larger effect on the course. From these judgments we obtain the priorities shown in Fig. 5. As an example of node comparisons, in Fig. 6 we give the judgment (the preference) for experienced versus recent invigilators on test based exams. It is more important to have an experienced invigilator on test based exams (because such exams provide the students more room to cheat and this type of exam needs more care to invigilate), we have entered the value three for experienced over recent for this comparison. As a result, experienced invigilators play a more significant role in their influence on the examination process. The priorities derived from the pairwise comparisons of the factors provide the essential inputs for the unweighted supermatrix while the priorities derived from the pairwise comparisons of the clusters are multiplied times the appropriate elements there to give the weighted supermatrix (Fig. 7). The limit of the weighted matrix is calculated to net out the dominance for all the factors in the system. Since the problem has different participants, a group consisting mostly of invigilators is asked for the comparison judgments and, as in the AHP, the geometric mean is used to combine the many judgments into a single representative judgment for that comparison. The limit matrix gives synthesized values that provide relative priorities of both the alternatives and the elements in the clusters. The unweighted, weighted and limit matrices are given in the Appendix. Table 1 lists the priorities from the limit matrix.
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Fig. 4. A screen view of paired comparisons of clusters.
Fig. 5. Priorities produced from the judgments in Fig. 4.
Fig. 6. A screen view on paired comparison of nodes.
Fig. 7. Priorities produced from the judgments in Fig. 6.
As can be seen in Table 1, for Saturday or open book type exams, the invigilators assigned at 9 am have greater weights. Similarly assignments to undesired times have greater importance than the remaining objectives. This result enables the decision maker to load exams scheduled at undesired time slots on the invigilators equally if they must be assigned to undesirable time slots. Similarly, experienced invigilators and providing required resources by the administration for the assignment process are important factors to consider.
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Table 1 Problem objectives and the criteria with their relative weightsa . Cluster
Element
Priority
Cluster
Element
Priority
Administration
The performance of the exam process Providing required resources for the assignment
0.081 0.128
Faculty
Performance of the supervising Requests on related exam
0.065 0.129
Invigilators
Experienced New Number of exams
0.051 0.025 0.011
Exam duration
60 min 90 min 120 min
0.007 0.012 0.021
Exam time slot
9–11 11–1 2–4 4–6
0.010 0.005 0.004 0.006
Exam day
Monday Tuesday Wednesday Thursday Friday Saturday
0.003 0.002 0.002 0.002 0.003 0.006
Exam type
Open book Test based Paper based Without answer sheet
0.011 0.010 0.007 0.005
Students
TA related invigilators
0.086
Alternatives
Min-max assignment to undesired time slots Min-max individual total loads Min total assignment cost
0.014 0.011 0.010
Course
Easy Difficult
0.099 0.173
a
Note that the largest weight in each cluster is italicized.
Table 2 Overall outcome. Alternatives
Limiting
Normalized priority (normalized by cluster)
Min-max assignment to undesired time slots Min-max individual total loads Min total assignment cost
0.014 0.011 0.010
0.40 0.31 0.29
The outcome of the study can be used in different ways to facilitate the exam scheduling process. As well as prioritizing the objectives we also obtain the priorities of all the factors defined for this problem. It is seen that the exams with 120 min duration or those scheduled on Saturday have greater importance as expected. Greater importance means greater effort, attention, cost or time spent during invigilation. Similarly, early morning, late afternoon or weekend exams, the ones in the long form, or the exams scheduled on the last day of the week have greater importance level because they need more effort to supervise. The normalized values for the Alternatives are shown in the second column in Table 2. A relative weight for each criterion within any cluster can be similarly obtained by normalizing the values in that cluster. The next section introduces the new solution after solving the mathematical model again by using the ANP outcome obtained here. The weights for each exam are in terms of its ANP priorities depending on its time slot, day, and duration. 4. Solution of the invigilator-exam assignment problem with the new weights The mathematical optimization model developed above was solved using the weights obtained from the ANP model. As an example of a similar parameter estimation study from the literature, Saaty et al. have shown that the coefficients of a mathematical linear programming (LP) model can be represented with priorities obtained with relative (i.e., pairwise comparisons) measurement. Intangibles such as effort and skill are not usually included directly in a mathematical model because of the absence of a unit of measurement. However, intangibles can be quantified through the relative measures or priorities of the AHP or ANP [13]. In our case, the invigilator-exam scheduling problem uses the priorities of intangibles such as objective priorities and exam weights obtained from an ANP analysis along with normalized measures of tangibles (when present) in a nonlinear optimization model with coefficients and variables. In passing we emphasize that the priority values range from zero for desirable to 1 for very undesirable. Thus the larger the priority is, the higher the load. As large priorities are undesirable outcomes in terms of ANP priorities need to be minimized. Table 3 gives the weights for each exam for its scheduled day, time slot, type of exam, duration, time of day and difficulty of its course. It also gives the number of invigilators required for each exam (gj ), the ordinal weight for the old method of scheduling (aj ) and the ANP summary weight.
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Table 3 Ordinal and ANP weights for difficulty of each exam. Date
Time
Course
gj
aj
P
# of invigilators
Ordinal weight
ANP weight
Time slot
Type
Duration Day
Difficulty
(exam component weights)
12.12.2005 Monday
09:00 09:00 11:00 11:00 14:00 16:00 16:00
Fizik I(A, B)a Physics I Yöneylem Araş. I (A, B) Operations Research I Mühendislik Malzemeleri (A, B) İnsan Kaynaklari Yön. Pazarlama Yön. (A, B)
2 1 2 1 2 1 1
2 2 3 3 2 2 2
0.032 0.032 0.036 0.036 0.021 0.026 0.026
0.010 0.010 0.005 0.005 0.004 0.006 0.006
0.007 0.007 0.007 0.007 0.007 0.005 0.01
0.012 0.012 0.021 0.021 0.007 0.012 0.007
0.003 0.003 0.003 0.003 0.003 0.003 0.003
0.099 0.099 0.173 0.173 0.099 0.099 0.099
12.13.2005 Tuesday
09:00 09:00 11:00 14:00 16:00 16:00
Matematik I (A, B) Calculus I İngilizce I (A and B) Benzetim (A, B) Diferansiyel Denk. (A, B) Differential Equ.
3 1 2 3 2 1
3 3 2 3 3 3
0.040 0.040 0.024 0.032 0.036 0.036
0.010 0.010 0.005 0.004 0.006 0.006
0.007 0.007 0.01 0.005 0.007 0.007
0.021 0.021 0.007 0.021 0.021 0.021
0.002 0.002 0.002 0.002 0.002 0.002
0.173 0.173 0.099 0.173 0.173 0.173
12.14.2005 Wednesday
09:00 11:00 14:00 14:00 18:00
Türk Dili I (A, B) Bilg. Prog. II (A, B) Teknik Resim (A, B) Mühendislik Ekonomisi (A, B) Üretim Planlama (A, B)
2 3 2 2 2
2 3 3 3 4
0.029 0.024 0.034 0.025 0.036
0.010 0.005 0.004 0.004 0.006
0.01 0.005 0.007 0.007 0.007
0.007 0.012 0.021 0.012 0.021
0.002 0.002 0.002 0.002 0.002
0.099 0.173 0.173 0.173 0.173
12.15.2005 Thursday
09:00 09:00 09:00 09:00 09:00 11:00 11:00 14:00 16:00
Kimya (A, B) Chemistry KKİT ESKA NKV (A, B) Olasılık (A, B) Probability (A) Üretim Yöntemleri (A, B) Teknik İngilizce III (A, B)
2 1 1 1 2 2 1 3 2
2 2 2 2 2 3 3 2 2
0.031 0.031 0.031 0.031 0.031 0.035 0.035 0.025 0.030
0.010 0.010 0.010 0.010 0.010 0.005 0.005 0.004 0.006
0.007 0.007 0.007 0.007 0.007 0.007 0.007 0.007 0.015
0.012 0.012 0.012 0.012 0.012 0.021 0.021 0.012 0.007
0.002 0.002 0.002 0.002 0.002 0.002 0.002 0.002 0.002
0.099 0.099 0.099 0.099 0.099 0.173 0.173 0.099 0.099
12.16.2005 Friday
09:00 09:00 11:00 14:00 16:00
TBTK (A, B) Teknik İngilizce V (A, B) Tesis Planlaması (A, B) İstatistik II (A1, A2, B1, B2) Mühendislik Mek. (A1-2, B1)
1 2 2 2 2
2 2 2 3 2
0.035 0.030 0.031 0.035 0.032
0.010 0.010 0.005 0.004 0.006
0.015 0.01 0.016 0.007 0.016
0.007 0.007 0.007 0.021 0.007
0.003 0.003 0.003 0.003 0.003
0.099 0.099 0.099 0.173 0.099
12.17.2005 Saturday
11:00 14:00
Sosyal Seçmeli I İş Etüdü (A1-2, B1)
2 2
3 4
0.025 0.038
0.005 0.004
0.007 0.007
0.007 0.021
0.006 0.006
0.173 0.173
a
(A, B) used to indicate more than one group for same course.
On the other hand, these ANP weights in Table 3 correspond to the sum of the weights of time slot, type, duration and the day. Though we get the importance of the course difficulty in this topic, we do not consider it as a criterion that affects an invigilator’s attitudes during his exam invigilation. It does not matter whether if that course is a difficult one or not since his duty is to invigilate the exam anyway. The time slot, the duration, day and the type of the exams are treated as the factors that affect the weight of an exam related to that course. This is how we calculated the exam weights to incorporate them the mathematical model. As an example, Operations Research I, scheduled at 11.00 am on Monday, is a paper based exam with the duration of which is 120 min. The weights for the specific characteristics of each exam are obtained from Table 1 and are given in the last 5 columns of Table 3. The values (except difficulty criterion) for the Operations Research I exam are then 0.005, 0.007, 0.021, and 0.003 and summing them yields an ANP weight of 0.036 for this exam. The number of invigilators is 1 and the old ordinal weight assigned to this exam would be 3. The ANP weight may be interpreted as the exam load from the invigilator’s point of view; the smaller the weight, the less the load. From the administration’s point the number of invigilators required is an important cost and enters into the assignment process during the optimization calculations. The ordinal weights used in the earlier work were assigned by administrators as their gestalt view of the exam, including the number of invigilators required. Since the exams are already scheduled, we are able to recalculate their weights as shown in Table 3 and the model is resolved. Table 4 gives two solutions. As shown in Table 5; with the new weights, we have r1 = 0.45 (the maximum individual load in the second column) while the corresponding r1 value for the solution without ANP weights is 0.44 (the maximum individual load, in the first column). This outcome shows that the maximum individual load r1 (0.44) without using the ANP is actually more (0.45) than if we use the ANP outcome to weight the exams. With the older manual way and also with the first model described
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Table 4 Previous and new invigilator-exam assignments. Exam
Weights Current
Fizik I (A, B), Physics I Yöney. Araş. I (A, B), OR Müh. Malz., (A, B) Insan Kaynakları Yön. Pazarlama Yön. (A, B) Math. I (A, B), Calculus Ingilizce I (A, B) Benzetim (A, B) Dif. Denk., Differential Equ. Türk Dili I Bilgis. Prog. II Teknik Resim Mühendislik Ekonomisi Uretim Planlama Kimya, Chemistry KKIT ESKA NKV (A, B) Olasılık, Probability (A) Uretim Yöntemleri (A, B) Teknik İngilizce III (A, B) TBTK (A, B) Teknik İngilizce V Tesis Planlaması (A, B) Istatistik II (A, B) Mühendislik Mekaniği (A, B) Sosyal Seçmeli I İş Etüdü (A, B)
New
2 3 2 2 2 3 2 3 3 2 3 3 3 4 2 2 2 2 3 2 2 2 2 2 3 2 3 4
0.032 0.036 0.021 0.026 0.026 0.04 0.024 0.032 0.036 0.029 0.024 0.034 0.025 0.036 0.031 0.031 0.031 0.031 0.035 0.025 0.03 0.035 0.03 0.031 0.035 0.032 0.025 0.038
Previous solution
Solution with ANP weights
Invigilators
Invigilators
1
2
3
4
1 1 1
1 1 1
1
1 1
5
6
7
8
1
1
1 1
9
1
1
1 1
2
3
4
1 1
1 1
1
1 1 1
1
1
1 1
1
1
1 1
1
1
1 1
1
1 1
1 1 1 1 1
1 1
1 1 1
1 1 1 1
1
1
1 1
1
1
1 1
1 1
1
1
7
8
1
1
9
1 1
1 1
1 1
6
1
1 1 1
5
1
1
1 1 1
1 1 1
1 1 1 1 1 1
1 1
1
1 1
1 1 1
1 1 1 1 1 1 1
1 1 1
1 1 1
1 1
1 1 1
1
1 1
1
1 1
1
1 1
1
1 1
1 1
1 1
1 1 1
1 1 1 1 1
1 1 1 1
1 1
1 1
1 1 1
1 1
1 1
1
1 1 1
1
1 1 1
1 1 1 1
1
1
1 1
1
1 1 1 1
1 1
1
1
1
1
1
1 1
1 1 1
1 1
1 1
1 1
1 1
1
1 1 1 1 1
1 1 1
1 1 1 1
1
1
1
1 1 1
1
1
Table 5 Invigilator loads for two solutions with and without ANP.
Without the ANP With ANP
I1
I2
I3
I4
I5
I6
I7
I8
I9
0.36 0.34
0.43 0.44
0.44 0.44
0.44 0.45
0.43 0.43
0.44 0.42
0.24 0.21
0.24 0.22
0.41 0.45
above we had used arbitrary numbers for the exam weights, but the outcome was not very reliable. With the ANP, we have greater confidence in the outcome because we estimate the exam and the weights of the objective functions by using a well-understood systematic method, the ANP. Actually as we mentioned before, we do not consider the difference between the optimum values as so crucial. Though the maximum loads are almost the same, there are still slight differences among the individual loads in two set of solutions. For example, invigilator 7 has 0.24 as his total load in the first solution while his load is 0.21 in the second one which means the second assignment is better for him. On the contrary, for invigilator 9, these values are 0.41 and 0.45 respectively and this shows a decline in his satisfaction as we explained before. We surmise that this second set of exam weights is obtained in a more realistic way having considered all the dependencies and feedback among the criteria and having used the ANP we have a better way to represent the real dynamics of the problem. As mentioned previously, invigilator assignments used to be done manually and the only criterion was to assign an equal number of exams to each invigilator. Since there was no consideration of exam loads in this process, there used to be big gaps and dissatisfactions among the persons, and the invigilators had already complained about being assigned exams equally. As we explained in Section 2, a mathematical model that considers these disparities gives a better solution in terms of individual satisfaction. The proposed approach, on the other hand, considers system dynamics, includes people’s expectations with all their inner and outer dependencies (in obtaining the model parameters), uses carefully calculated system parameters and gives a better understanding of the educational system as well as increasing the confidence and overall benefits to the organization.
940
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5. Conclusion Mathematical models are important tools for optimization problems. The quality of the solution of such a model depends strongly on the estimated values of the parameters of the problem. In this study an important – but not well studied part – of an educational scheduling problem is solved after estimating its parameters by using an ANP model. All educational institutions have such issues: exams, courses, students, instructors, invigilators and the administration. Despite the fact that the judgments related to them may change according to the specific considerations and constraints, the ANP model developed here can provide a general framework to obtain the relative weights of similar assignment problems in other institutions. Appendix Unweighted matrix
Weighted matrix
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Limiting matrix
References [1] M.W. Carter, G. Laporte, Recent developments in practical course timetabling, in: Selected Papers from the Second International Conference on the Practice and Theory of Automated Timetabling II, in: Lecture Notes in Computer Science, vol. 1408, 1998, pp. 3–19. [2] S.M. Al-Yakoob, H.D. Sherali, M. Al-Jazzaf, A mixed-integer mathematical modeling approach to exam timetabling, Computational Management Science (2007) doi:10.1007/s10287-007-0066-8. [3] P. Cowling, G. Kendall, N.M. Hussin, A survey and case study of practical examination timetabling problems, in: Proceedings of the 4th International Conference on the Practice and Theory of Automated Timetabling, PATAT2002, 2002, pp. 258–261. [4] A. Schaerf, A survey of automated timetabling, Artificial Intelligence Review 13 (2) (1999). [5] G.M. White, B.S. Xie, S. Zonjic, Using tabu search with longer-term memory and relaxation to create examination timetables, European Journal of Operational Research 153 (2004) 80–91. doi:10.1016/S0377-2217(03)00100-0. [6] Z.K. Ozturk, G. Ozturk, M. Sagir, An automated multi-objective invigilator-exam assignment system, International Journal of Information Technology and Decision Making 9 (2) (2010). [7] T. Chang, N. Meade, J. Beasley, Y. Sharaiha, Heuristics for cardinality constrained portfolio optimisation, Computers and Operations Research 27 (2000). [8] T.L. Saaty, Fundamentals of Decision Making and Priority Theory with the Analytic Hierarchy Process, in: AHP Series, vol. VI, RWS Publications, Pittsburgh, 2000. [9] N. Ahmad, Daniel B, G.R. Simons, The integration of analytical hierarchy process and data envelopment analysis in a multi-criteria decision-making problem, International Journal of Information Technology and Decision Making 5 (2) (2006). [10] P. Aragone, J. Aznar, Valuation of urban industrial land: an analytic network process approach, European Journal of Operational Research 185 (2008). [11] E.E. Karsak, S. Sozer, S.E. Alptekin, Product planning in quality function deployment using a combined analytic network process and goal programming approach, Computers & Industrial Engineering 44 (2002) 171–190. [12] T.L. Saaty, Decision Making with Dependence and Feedback The Analytic Network Process, RWS Publications, Pittsburgh, PA, 2001. [13] T.L. Saaty, L.G. Vargas, K. Dellmann, The allocation of intangible resources: the analytic hierarchy process and linear programming, Socio-Economic Planning Sciences 37 (2003) 169–184.
