Problem Title: University Course Scheduling

Problem Description:

Your task is to create an optimal class schedule for a university's upcoming semester. The university has three departments: Science, Engineering, and Humanities. Each department offers a set of courses, and your goal is to schedule them to meet various constraints while maximizing room utilization and minimizing empty time slots.

Problem Details:

Department and Courses:

Science Department $($SD$)$: Offers $6$ courses $($S$1,$ S$2,$ S$3,$ S$4,$ S$5,$ S$6).$

Engineering Department $($ED$)$: Offers $4$ courses $($E$1,$ E$2,$ E$3,$ E$4).$

Humanities Department $($HD$)$: Offers $5$ courses $($H$1,$ H$2,$ H$3,$ H$4,$ H$5).$

Room Availability:

There are $5$ available classrooms $($A$,$ B$,$ C$,$ D$,$ E$).$

Each classroom has specific time slots when it is available, as shown in Figure $1.$

Course Constraints:

Each course has specific preferences for its time slot $($see Figure $2).$

No two courses from the same department can be scheduled at the same time.

Courses cannot overlap in the same room.

Instructor Preferences:

Some instructors have preferences for teaching specific courses at certain times $($see Figure $3).$

Room Capacity:

Each classroom has a maximum capacity, as shown in Figure $4.$