Developing an efficient optimization of course-lecturer distribution timetabling using transportation algorithm: A case study of Federal Polytechnic Offa

The institution course timetabling problem (ICTP) is a multidimensional assignment-problem that varies from course timetabling, class-teacher timetabling, student scheduling, teacher assignment, and classroom assignment. Many researchers have attempted to solve problems as related to timeslot but neglecting areas of course allocation to lecturers. The paper presented a course allocation and distribution model for lecturers based on their fields of interest and qualification to a transportation algorithm which was aimed at optimising the performance of lecturers in each course. It also evaluated overall efficiency of lecturers without exceeding the maximum workload. The performance of the course-to-lecturer allocation of the electrical/electronic engineering department, federal Polytechnic Offa, Kwara State was collected using simple questionnaire. The information obtained from the questionnaire was used to test the Algorithm developed. The result showed that using the developed algorithm for course distribution, the performance is 76.98% and 82.1% for the first and second semesters respectively. This showed that using the algorithm for allocation of courses to the lecturers of any department can be done based on input data without exceeding the recommended workloads of each cadre. This improved the quality of teaching, save time, and resources compare with manual methods. The study therefore recommended that future work should include practical distribution among technologists, sharing the excess workload to a particular lecturing grade as the case may be.


Introduction
The research on timetabling problems has a long history and can be dated back to the last sixty years. Before 1980, various attempts had been made to address the issue timetabling problems. Numerous researchers have studied Timetabling as a complete Network Problem (NP), Even, Itai, and Shamir (1976) & Garey and Johnson (1979). Timetabling problem has received special attention of the scientific community over the years. For instance, a biennial conference on the Practice and Theory of Automated Timetabling (PATAT) has been going on since 1995. Similarly, both Association of European Operational Research Societies (AEORS), and Working Group on Automated Timetabling (WATT) were established in 2002.
Timetabling process can be divided into two phases. The first phase is the curriculum definition of each class of students and assigning of teachers to courses. The second phase is scheduling of the curriculum courses into time slot, based on the available resources such as manpower, and equipment to the classes which is compatible with the entire previously defined requirement. Abramson and Abela, (1991), Erben and Keppler (1995), Herz (1992), Monfroglio (1988) Paechter, Rankin, Cumming, and Fogarty (1998), Ross, Hart and Corne, (1994), Schaerf (1996), Carter and Laporte (1998), De Werra (1985), Schaerf (1999)   propose methods used in solving the timetabling problems. Carter and Laporte, (1998) present the major differences between different methods of solving the course timetabling problems at the school level, such as that of high school and also at the university level. According to them, the institution course timetabling problem (ICTP) is a multi-dimensional assignment problem, in which students and lecturers are assigned to courses, venues and time slots.
On the first phase, Andrew and Collins (1971) proposed a procedure for assigning the teachers to courses, based on a simple linear programming technique with some limitations pointed out by Tillett, (1975), while Breslaw, (1976) propose a model to overcome the limitation of the model proposed by Tillett (1975). Schaerf, (1996) applied a tabu search to solve the school timetabling problem for an Italian school. Randall, Abramson, and Wild (1999) has also used a tabu search to solve the Abramson set of school timetabling problems. Harwood and Lawless (1975) used a goal programming model to solve the teacher assignment problem. Liu, Zhang and Leung (2009) & Randall, Abramson and Wild (1999) have also successfully applied simulated annealing to the school timetabling problem. Schniederjans and Kim (1987) highlighted the drawbacks in implementing this model and proposed a model to overcome it and also mentioned some factors that could affect the size and complexity of the teacher assignment problem.  (2002) proposed a hybrid approach for solving the final examination timetabling problem that generates an initial feasible timetable using constraint programming, and then applied simulated annealing with hill climbing to obtain a better solution.
Beligiannis and Tassopoulos (2012) solved the Greek school timetabling problem using particle swarm optimization. The particle swarm optimization produced better timetables than other techniques such as evolutionary algorithms and constraint programming. Burke and Bykov, (2008) proposed a general and fast adaptive method that arranges the heuristic to be used for ordering examinations. Turabieh and Abdullah (2009) proposed an electromagnetism-like mechanism with force decay rate great deluge algorithm for university course timetabling which is based on an attraction-repulsion movement for solutions in the search space. All these focused on the second phase of timetabling problem. From the literature, very few works pay attention to the performance of course-to-lecturer problems where the bulk of the performance efficiency of the students lies; the aspect of the professional degree required for teaching a particular course of all lecturers and the workload of the lecturers. The present paper addresses the aspect of assigning courses to measure the performance of curriculum management and course-to-lecturer efficiency. The paper specifically focuses on the transportation algorithms modelling and optimization of the algorithm.

Transportation Algorithms Modelling
This involves determining an optimal strategy to maximize quality of teaching and performance of lectures through allocating to them courses that are relevant to their professional qualification and their field of interest or study without exceeding the maximum workload. In this case, the courses have their credit units, the lecturer have their maximum workload depending on their cadre, level and other responsibilities. The courses are regarded as sources to various lecturers called destination. The problem to be solved is how to maximize the efficiency of courses allocated with performance of lecturers. Each lecturer has a fixed workload of the credit units usually called capacity or availability.

Definition of Notation
P ij represents the percentage of professional degree that lecturer i can teach a course j x ij represents the number of credit unit to be distributed from course j to lecturer i (i= 1,2,3,…,n; j=1,2,3,…,m)  The optimum performance is given as: To maximize the performance, the problem can be rewrite as follow: The objective function is given as

Algorithm optimization
The total number of courses taken by the student can be divided into two. These are the departmental courses taught by the lecturers of the departments and external courses taught by the lecturers of other departments. Our focus is on the departmental courses which can be categorised into three options; i.e. computer, control and instrumentation; electronics and telecommunication; and power and machine. The problem of assigning of lecturers to the courses and course sections are in 2 phases.

Phase 1
The first phase is allocating lecturers to the courses base on their professional options and skills, and determining the number of courses to be assigned to each lecturer. The Second phase is, scheduling the teacher to the course sections in order to balance the teachers' workload.

Phase 2: Algorithm Optimization
Feasible Solution using North-west corner method and Stepping-stone method Step 1: arrange the data into row (lecturers) in descending order of grade levels and column (courses) Step 2: start by selecting the cell in the North-west corner of the table Step 3: assign the maximum credit unit of course in this cell, based on the requirements and lecturer and course constraints Step 4: Exhaust the credit load from each lecturer before moving to the next lecturer Step 5: exhaust the credit load of each column before moving to the next column Step 6: check if total credit units is equal to total credit load Step 7: if yes, select an unused square box to be evaluated Step 8: begin from at the unused square box to trace a close-path back to the original by moving horizontally or vertically only.
Step 9: Beginning with positive sign (+) at the unused square box by placing alternatively minus sign (-) in the next square box follow by plus sign (+) to each corners square box of the traced close path Step 10: evaluate an improvement index, I ij by adding the performance found in each square box containing a plus sign and the subtracting the units performance in each square box containing minus sign (-) and divide all by total credit unit Step 11: go back to step 7 until an improvement index, I ij has been calculated for all the unused square box Step 12: check if all I ij > 0, an optimal solution has been reached Step 13: if I ij < 0, it is possible to improve the current solution and increase total performance Step 14: choose the closed-path with I ij < 0 Step 15: select the minimum credit unit in the closed path and add or subtract from each cell using the assign signs Step 16: go back to step 7 Step 17: print the lecturers name, assigned courses and credits, percentage of performance and optimum performance.  However, in the first semester, HND seminar which is 1 credit unit will be supervised by all lecturers from Lecturer I and above while in the second semester, HND project which is 4 credit units will be supervised by all lecturers from lecturer I. The ND project which is 4 credit units will be supervised by all lecturers from Lecturer III and above. This will not be input since it involves almost all the lecturer taking a particular course.
For the first semester, the total credit load = 390, Index ratio = 0.251, and for the second semester, index ratio = 0.236. The new credit load is displayed in fig. 4

Figure 4
The adjusted maximum Credit load per Grade level for the 2 semesters The output of transportation algorithm is summarised in the Fig. 5

Findings and Conclusion
Based on the instruments employed in this study, it was discovered that the performance of the lecturers in any course after the course allocation is not less than 60% and the overall % performance for electrical engineering courses if all the courses are assigned according to the output of the programme is 76.98%for the first semester and 82.1% for the second semester base on the input performance of all lecturers in each course. It was also found out that any variation in the input performance of lecturer in a course will result into new assignment of courses and new % level of performance. With this algorithm, this study concluded that the course allocation of any department can be done based on the input data of lecturers and courses without overloading any lecturer and with the best performance combination. The course allocation and distribution can be modelled into transportation and assignment problems and solved using other linear algorithms. We believe that the performance of Transportation algorithm for the course allocation and distribution can be improved by applying advanced assignment operators and heuristics.