Experimental Analysis

Table #table01#508> lists all major components of the data and Tables #table02#509> and {#table03#510> show the rates of sparseness of the three-semester data which greatly depends on number of time slots and rooms used. The level of sparseness increases as the overall size of the data, particularly, number of rooms, time slots, and number of classes. Also, after scheduling all the classes, there will be about #tex2html_wrap_inline1868# spare space-time slots, hence, the sparseness of the problem is defined as as the ratio #tex2html_wrap_inline1870#. As we can see that Table #table03#513> degree of sparseness is less than that of Table #table02#514> and the reason is the presence of the students' preferences. Our overall results are shown in Tables (#table1#515>, #table2#516>, #table3#517>, #table31#518>, #table4#519> and #table41#520>). Table #table1#521> gives a percentage output of the preprocessor (the expert system) for a three-semester set of data as well as the highest and lowest percentage of scheduled data. We also can see that the size of those scheduled classes increases as the input (number of classes) decreases, which was not really surprising since the overall number of rooms and time slots were kept fixed for the three semesters. So the system does quite well for small sets of classes even when both class and students constraints were taken into consideration. Unlike Table #table1#522>, Table #table2#523> shows improved results and that was mainly due to not taking students constraints into consideration and only restricting the system to class constraints. In the case of simulated annealing, Table #table3#524> shows output in a percentage form averaged over 10 runs for three semesters without a preprocessor. Clearly, SA using the geometric annealing schedule shows a rather poor result which is even lower in quality to that of the expert system of Table #table1#525>. Also, the use of the adaptive and the cost-based schedules did not improve the results by much and remained below those of Table #table1#526>. Furthermore, the cost-based reheating schedule gave an overall better average result than that of the geometric and the adaptive schedules for the three semester. In contrast to the the numbers in Table #table3#527>, Table #table31#528> shows higher percentages of the three annealing schedules and it is mainly due to the exclusion of students' preferences/constraints and only dealing with class constraints. Table #table4#529> shows quite improved and high quality averaged results for the three schedules. Such excellent results can only be attributed to the use of preprocessed input (output of the rule-based preprocessor) Also the figures were averaged over 10 runs of annealing and for the cost-based schedule all those runs for the third semester set yielded a perfect schedule, that is, all classes of the input set were scheduled satisfying the given class and students constraints. However, as we can see in Table #table01#530> that the size of classes of the third semester is quite small relative to the first two semesters so achieving a perfect schedule for it was not too difficult. Furthermore, we obtained even better results for the three semesters when the class constraints were disjoined from the students preferences and first dealt with the class constraints. As shown in Table #table41#531> we achieved perfect schedules for the three semesters using the cost-based reheating schedule and also a perfect schedule for the third semester using the adaptive annealing. Students data and preferences are dealt with in the second stage with respect to the class schedule obtained in the first stage. Clearly, this tells us that the use of preprocessing was not only helpful but quite essential when using methods such as simulated annealing to deal with complex timetabling problems. Furthermore, these high quality averaged results state in an unambiguous terms that a multi-phase approach, one as explained above, gives a much better results to timetabling problems then a single-phase approach.