# Allocating Students to Industry Placements Using Integer Programming and Ant Colony Optimisation
## 7 Experiments and Results
### Results  
Tables 4 and 5 present the results of the IP model and \(\mathcal{MM}\)AS on problem instances generated by the various distributions discussed in Section 6. The first two columns specify the number of students (\(Stu.\)) and companies (\(Comp.\)), respectively. The next four columns specify the criteria used to measure the quality of solutions (three columns) and the time taken (in seconds) by the IP. The first is \(\#1\)\(s\), or the number of students assigned to companies with a preference rating of 1. The larger the number of students with \(\#1\)\(s\), the better the solution. Second, \(Avg.\) or the average allocation of student preferences assigned to the companies in a solution. Third, \(Obj\) is objective value in Equation (2). The next six columns are associated with \(\mathcal{MM}\)AS. Since 25 runs per instance with \(\mathcal{MM}\)AS were conducted, standard deviations (SDs) associated with each criteria are also shown. For most instances (up to 3000 students and 3000 placements), the IP is able to find optimal solutions, often quite quickly. The IP solutions are better than the \(\mathcal{MM}\)AS solutions considering number \(\#1\) preferences and average allocations for all these instances. When the proportion of students to companies is high (25% and 50%), solutions with students having a preference of 1 are found for all placements. Beyond 3000 students, the IP struggles to find solutions within the 30 min time limit. In these cases, \(\mathcal{MM}\)AS finds solutions, but it is not possible to quantify their quality. Despite no guarantee on the quality of the solution, at the very least a feasible solution to any large instance can be found within five minutes. For real problems, these solutions can be provided to the industry partners in lieu of a manual allocation that can be very problematic in construction. Not surprisingly, the IP run-times progressively increase as the problem size increases. Runs take more than a minute when there are more than 1000 students and 750 companies. However, this is still a feasible time-frame to obtain good solutions. As previously discussed, \(\mathcal{MM}\)AS provides an alternative when the IP run-times become too large. This leads to a distinct advantage in real student allocation systems, where solutions can be built, modified (e.g., incorporating a constraint such as a student must not be allocated to a company), and re-built quickly if the user is unhappy with the allocation for some reason. The objective criteria are investigated further in Figures 2-4, using data with at most 2500 students. Larger instances are not considered since the IP model often fails to find solutions within the time limit. Figure 2 shows the difference in #1 preferences from \(\mathcal{MM}\)AS to the IP Model (\(\frac{\mathcal{MM}AS-IP}{IP}\)). The chart on the left splits the results by student numbers, whereas the figure on the right splits the results by the proportion of students to placements ([0.25, 0.5, 0.75, 1.0]). Firstly, regarding the number of students, we see that the differences increase with increasing problem size, i.e., \(\mathcal{MM}\)AS does not scale well. However, once the problem instances are very large (\(>\)2000 students) the advantage of the IP model is diminished. In fact, as it is seen in Table 5, if the problem instances are very large, \(\mathcal{MM}\)AS is by far the preferred method since on occasion the IP model struggles to even find solutions. On examining the chart to the right of Figure 2, we see that in the datasets with low proportions (0.25), the differences between \(\mathcal{MM}\)AS and the IP model are large. The difference reduces with increasing proportions (0.5 and 0.75) followed by a slight increase when the proportion is 1.0. This is not surprising since when the proportions of placements are low, the IP model is significantly more efficient and is able to find the optimal solutions.