Social distancing during COVID-19 has become one of the most important measures in reducing the risks of the spread of the virus. Implementing these measures at universities is crucial and directly related to the physical attendance of the populations of students, professors, employees, and other members on campus. This research proposes an automated scheduling approach that can help universities and schools comply with the social distancing regulations by providing assistance in avoiding huge assemblages of people. Furthermore, this paper proposes a novel course timetable-scheduling scheme based on four main constraints. First, a distance of two meters must be maintained between each student inside the classroom. Second, no classrooms should contain more than 20% of their regular capacity. Third, there would be no back-to-back classes. Lastly, no lectures should be held simultaneously in adjacent classrooms. The proposed approach was implemented using a variable neighborhood search (VNS) approach with an adaptive neighborhood structure (AD-NS) to resolve the problem of scheduling course timetables at Al-Ahlyyia Amman University. However, the experimental results show that the proposed techniques outperformed the standard VNS tested on university course timetabling benchmark dataset ITC2007-Track3. Meanwhile, the approach was tested using datasets collected from the faculty of information technology at Al-Ahlyyia Amman University (Jordan). Where the results showed that, the proposed technique could help educational institutes to resume their regular operations while complying with the social distancing guidelines.

In operation research (OR) the problem is modelled quantitatively, whilst in artificial intelligence (AI) it is modelled as a graph (i.e., tree, network or others). After the problem is modelled, we need to solve the problem by exploring different solutions in order to choose the best one among them (search algorithm). The optimality consists of two types of solutions: global and local optimal solutions. The local optimal is the best solution among others within search space region, but not all regions in the search space. Whereas the global optimal is the best solution among all solutions in search space (all regions). In order to find the good local optimal solutions/probably global optimum, an effective search mechanism is needed to help in explore promising regions in the search space. In every search process, there are three common phases: (i) the initial solution (initial start or point); (ii) technique to generate new solutions; (iii) termination criteria. There are three types of search mechanisms [

Analytical Search: It is used a mathematical function as a guide to find the optimal solution if possible (exist). By using this search mechanism there is no greatness to reach the optimal solution in every case, but the local optimum.

Blind Search: It is also called the unguided search, where the search space is enumerated to search for the optimal solution exhaustively, and there is no guarantee to find the optimal solution.

Heuristic Search: also called a guided search and there is no guarantee to find the optimal solution same as other search types, but it can produce satisfactory solutions/high quality solutions in most cases. This thesis focuses on heuristic search.

The author in [

The work in [

Scheduling problem such as university course timetabling (UCTT) is considered as a non-deterministic polynomial (NP) complete problem. Many algorithms were implemented to accomplish scheduling tasks effectively. Reference [

According to [

University timetabling, in general, is similar for all universities. For instance, classes reside at the same level and almost identical in sizes. Moreover, each class holds a different group of students as students enroll in different courses and lectures. Work [

Hard constraints: For instance, two or more courses should be scheduled in different classrooms and timeslots.

Soft constraints: For example, lecturers with a majority of females are preferred to be assigned morning timeslots.

Roughly, the quality of a timetable is evaluated through such soft constraints and hard constraints. Soft constraints should reach a reasonable satisfaction degree, whereas hard constraints should reach a degree of completeness. For any constraint violation, the design will be penalized. So, the higher the penalties, the lower is the quality of the timetable design. Authors in [_{1,} _{2}, _{.}_{n}_{1,} _{2}, _{.}_{k}_{1,} _{2}, _{u}_{i}, there are _{i} _{rk} represents the set of the courses, such that CR = {_{r1}, _{r2}, _{rs}}

All previous works related to timetabling had focused on factors not related to social distance as a critical issue. However, new circumstances developed raised due to the perspective of a disease—COVID-19. COVID-19 is a pandemic that led to the worldwide practice of social distancing to minimize the spread of this virus. Universities and schools are two of the most crowded places, and mass gatherings at those places may cause the virus if social distancing is not maintained. This work provides a new approach to help universities schedule their courses in the manner of keeping students safe as much as possible. For this purpose, a new real-world dataset was collected from the faculty of information technology at Al-Ahlyyia Amman University (AAU). The Faculty of Information Technology has three departments and 12 curricula of students as shown in

#Days | #Courses | #Lectures | #Rooms | #Groups a |
---|---|---|---|---|

50 | 59 | 83 | 15 | 12 |

The problem of course timetabling is a combinatorial optimization problem [

Metaheuristics algorithms such as SA, TS, or VNS. However, VNS was used by [

Reference | Mechanism |
---|---|

[ |
Presented a hybrid Variable Neighborhood with Tabu Search (VNTS) approach for the median cycle problem, and the experimental results showed how to apply VNTS to any optimization problem. |

[ |
Proposed a hybrid algorithm of low-level hybridization between memetic algorithms and VNS. The achieved experimental results show the suitability of the approach and could find good optimal solutions. Moreover, on a set of standard test problems, new best-known solutions were produced for several instances. |

[ |
Presented a hybrid VNS with exponential Monte Carlo (EMC) acceptance criteria for the timetabling problem. However, the experimental results proved that the proposed approach outperformed the standard VNS and gave a comparable result among other approaches from the literature, and provided a suitable quality solution for small problems. |

[ |
Presented investigation research for several VNS approaches for solving the problem of university examination timetabling. They examined deferent neighborhood structures and introduced deferent initialization strategies such as greedy and random constructive, including a biased VNS and hybrid using decent VNS with a GA. Thus, the proposed technique was tested for wide benchmark problem instances. |

As shown in

It is possible to note that there are several neighborhood combinations such as union neighborhood structures [

Even though the employment of several neighborhoods will produce enhanced results, the computational time might be increased and searching capability may become limited. In this paper, a new VNS is introduced in combination with the adaptive neighborhood structure (AD-NS) that proposed by [

CBCT is based on scheduling all lectures according to a weekly schedule. Every lecture and a corresponding classroom for it must be set based on a given set of constraints (hard and soft constraints). Therefore, our approach is based on the following entities: weekday, timeslots, and periods. The weekday (day) entity describes five days of the week (Sunday to Friday). The timeslot entity is assigned to the teaching days (Sundays, Tuesdays, and Thursdays) when lectures last for 60 min, whereas lectures last for 90 min for Mondays and Wednesdays. Meanwhile, the algebraic product of timeslots and days generates the total number of scheduling periods, considering that each course has a fixed number of lectures to be scheduled in distinct periods, taught by a specific teacher, and attended by a certain number of students. Furthermore, the lectures of each course must have a minimum number of days to be spread around, considering that there are some periods in which the course cannot be scheduled.

When designing an approach for scheduling a timetable, we need to keep in mind that each room has a capacity that is equal to the total number of seats inside that room. Accordingly, for a large-enough room, all courses can be equally assigned while abiding by the COVID-19 social distance guidelines. In order to enforce social distancing on course scheduling, we need to handle the constraints mentioned in [

As mentioned above, there are four common hard constraints, which are as follows:

H1: For every course, lectures must be scheduled for distinctive periods.

H2: Two lectures cannot be assigned to the same period and the same room.

H3: Lectures of courses in the same curriculum or taught by the same instructor cannot be scheduled in the same period.

H4: If the teacher of a course is not available at a given period, then no lectures of the course can be assigned to that period.

In this section, we will present the seven soft constraints (S1–S7),

Soft constraint description | Number of violations counted | Proposed by | |
---|---|---|---|

S1 | For each lecture, the number of students attending the course must not exceed the capacity of the room where the lecture is held | V1 = 2 violations | [ |

S2 | Course lectures must be spread out over a specified minimum number of days. | V2 = 2 violations | [ |

S3 | All course lectures must be scheduled in one room; otherwise, the number of rooms occupied should be reduced. | V3 = 2 violations | [ |

S4 | (Curriculum distribution): For a given curriculum, V4 is counted if there is one lecture adjacent to any other lecture belonging to the same curriculum on the same day, which means that the students’ agenda should be as distributed as possible. | V4 = 2 violations | Proposed in this work |

S5 | (Room availability): For each timeslot, avoid assigning lectures in adjacent rooms. V5 is counted for every two lectures scheduled in two adjacent rooms at the same timeslot. | V5 = 8 violations | Proposed in this work |

S6 | (Building capacity): For each time slot, the occupation of the building should not exceed 20% of the building capacity. | V6 = 10 violations | Proposed in this work |

S7 | (Consecutive lectures): Avoid consecutive lectures in any room. | V7 = 10 violations | Proposed in this work |

Concerning the violations calculated and determined in

The VNS starts from the first neighborhood and moves on to the next one in sequence. Many local search methods can be applied within the VNS, including the descent method such as the VND. The VND uses a descent method to move around the neighborhood structures. In this paper, the changing neighborhood structure will be applied adaptively, as per the procedures described as follow [

Randomly select one type of soft constraint (i.e., curriculum compactness).

Order the lectures according to the selected soft constraint violations in a non-increasing way.

Select the lecture with the highest penalties Li, where Li belongs to Crk curricula (

Select another lecture Lj randomly that belongs to different curricula than Li (

Swap Li and Lj.

The procedures show diverting the search to other solution space and randomly swapping the lectures with the highest penalties with other lectures (free clash).

In this phase, the feasible initial solution is built by satisfying all the hard constraints (H1–H4) using a sequential greedy heuristic.

In this work, the VNS with adaptive neighborhood structure is proposed as shown in Algorithm.1. Thus, the approach starts with a given initial solution (Sol*) using the constructive algorithm; then, we set the initial non-improvement iterations to one (Non_improve = 1) to make sure that the solution avoids being trapped in a local optimum. After that, k neighbors are generated from N neighborhood structures by applying the AD-NS to select the best quality solution (Sol). Then the solution (Sol*) is accepted as the objective value only if its quality is better than the current solution (Sol). Otherwise, the non-improving iterations would be updated.

To achieve the highest fair percentage of comparison and to verify the efficiency of the algorithm, the study compared the proposed algorithm with standard VNS tested on university course timetabling benchmark dataset ITC2007-Track3.

The parameter settings for both (VNS-AD and VNS) are listed in

Parameters | VNS-AD-NS | VNS |
---|---|---|

Non-improvement iterations | Adaptively | Nothing |

Neighborhood’s structure | AD-NS | NS1 and NS2 |

Termination condition | 460 s | 460 s |

Where,

NS1: Moving a lecture from the current period to another free period.

NS2: Swapping lectures randomly.

^{3} benchmark datasets.

6 | 5.34 | 0.48 | 6 | 15 | 10.5 | 2.23 | ||

61 | 51.46 | 6.034 | 51 | 72 | 62.38 | 6.67 | ||

102 | 91.15 | 3.86 | 94 | 111 | 101.73 | 4.70 | ||

70 | 62.54 | 5.50 | 60 | 80 | 72.85 | 6.06 | ||

318 | 310.19 | 5.00 | 310 | 330 | 319 | 5.25 | ||

83 | 75.31 | 5.47 | 76 | 99 | 88.191 | 7.13 | ||

38 | 30.77 | 3.73 | 34 | 50 | 42.77 | 4.47 | ||

63 | 53.54 | 4.75 | 57 | 72 | 65.5 | 5.18 |

The experimental results in

Furthermore, the proposed approach in this work was tested on the data allocated for the Faculty of Information Technology at Al-Ahliyya Amman University (AAU). The collected dataset is described in

The comparison results between VNS_AD and the standard VNS is shown in

VNS-AD-NS | VNS | VNS-AD-NS |
|||||||
---|---|---|---|---|---|---|---|---|---|

Instances | B | W | M | STD | B | W | M | STD | P-value |

IT | 50 | 97 | 62.3 | 8.21 | 70 | 130 | 98.8 | 19.94 | 0.000 |

Where,

B: Best result, W: Worst result, M: Mean, STD: Standard deviation.

The achieved results of the proposed approach are compared with those of the standard VNS and the common soft constraints presented by ITC 2007, which is by given the calculation of social distancing between classrooms, students, and overcrowding buildings during academic lectures through the following proposed measurement as illustrated in

S = {x | x is the numbers of students attending lectures at level

R = {r | r is the room capacity at level i}

where β is the percentage students’ number per day _{m}, which is achieved using our approach in comparison with the actual number of students or actual room capacity R.

After 31 runs, the proposed approach shows that the result of ß is equal to 26.2354%, whereas the standard approach achieved 67.2022% in the case of social distancing for the academic schedule during the university day.

Social distancing is the main requirement during the time of COVID-19 for preventing the spread of the afflicting disease. Universities are forced to implement social distancing to prevent the spread of coronavirus among the students, teachers, and other university-related personnel since intensive groupings are one of the most common causes of rapid viral infection transmission. This work presents an approach to distribute lectures and class capacity to reduce massive student gatherings. It proposes new soft and hard constraints that would help reduce the percentage level of students per timeslot to the level suggested by health-care authorities. The soft constraints used in this work enhanced the distribution of students in a manner to avoid the high percentage of student numbers per day. Thus, the proposed approach achieved results that would reduce student crowding as compared to the standard approach, with an occupation of less than 26% of the actual university capacity while assigning timeslots. The present work can be implemented in all faculties and universities in Jordan or elsewhere to meet the conditions proposed by Al-Ahliyya Amman University.

This study guarantees a good result that ensure social distancing between students and employees inside the university campus. This will help to open universities to students and reduce virus infection. However, using Adaptive neighborhood structure will Leeds the VNS to select better neighborhood structure and divert the solution structure to a good promising region.

We would like to thank the Al-Ahlyyia Amman University for facilitating and funding this work.