Analyzing WMDD on Parallel Machines. Harrisen Egly, Alyssa Estrada, Kelcie Gerson, Sarah Murray
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1 Analyzing WMDD on Parallel Machines Harrisen Egly, Alyssa Estrada, Kelcie Gerson, Sarah Murray
2 Agenda Problem Setup: Determining final exams, classrooms, and a unique timeline Algorithm Overview: What is WMDD and how does it apply to this scenario? Four cases: How well does WMDD perform? When does it start to breakdown? Final Conclusions
3 Problem Setup Problem: P r j, d j Σw j T j Scenario: 40 IEOR classes that are assumed to have final exams, where each final exam is a job Ideal exam time determined from chart, as registrar would do Setup unique timeline Determine release time and due date from the ideal exam time and unique timeline Job weight is size of classroom Class Time Finals Time 8:40 am-9:55 am 9:00 am-12:00 pm 9:00 am-11:30 am 9:00 am-12:00 pm 10:10 am-11:25 am 9:00 am-12:00 pm 10:10 am-12:40 pm 9:00 am-12:00 pm 11:40 pm-12:55 pm 9:00 am-12:00 pm 1:10 pm-2:25 pm 1:00 pm-4:00 pm 1:10 pm-3:40 pm 1:00 pm-4:00 pm 2:40 pm-3:55 pm 1:00 pm-4:00 pm 4:10 pm-5:25 pm 4:00 pm-7:00 pm 4:10 pm-6:40 pm 4:00 pm-7:00 pm 5:40 pm-6:55pm 4:00 pm-7:00 pm 7:10 pm-9:40pm 7:00 pm-10:00pm
4 Problem Setup: Adjusting Our Timeline
5 Problem Setup: Adjusting Our Timeline
6 Problem Setup We decided to run the algorithm in 3 Classrooms 2 classrooms provided an uninteresting problem, as it requires putting one job in every slot (40 jobs in 40 slots) Schedule density: This metric allows us to compare the difficulty of scheduling jobs among the problems we encounter. It is calculated as: Schedule density measures the percentage of time spaces that will be non-idle in the final schedule
7 Algorithm Overview: Weighted Modified Due Date Weighted Modified Due Date (WMDD) is a greedy heuristic used to solved 1 r j, d j Σw j T j A combination of WSPT and Weighted Remaining Allowance Proved to be 1.06*OPT Class sizes were used as weights; this prioritizes classes whose lateness would require more individual rescheduling How It Works The Algorithm iterates over time units, calculating the modified due date as: On each iteration, the smallest (3 smallest) WMDD are added to the schedule
8 Case 1: Base Case In this case all 40 jobs are processed on 3 Machines with no additional constraints Major Assumptions: All Classes will fit in the available rooms No consideration of Major Tracks In this case, WMDD provides the optimal solution.
9 Case 1 Schedule Number of Tardy Jobs: 1 Objective Function: 20 Percentage of Tardy Jobs: 2.5% Schedule Density: 66.7% Time Exam Slot Mudd 303 Mudd 545 Mudd 833 Mon 9 am 0 IEOR 4709 IEOR 3609 IEOR 3402 Wed 9am 1 IEOR 4418 IEOR 4573 IDLE Mon 1pm 2 IEOR 4307 IEOR IEOR 4004 Wed 1pm 3 IDLE IDLE IDLE Mon 4pm 4 IEOR 4404 IEOR 4700 IEOR Wed 4pm 5 IEOR 6614 IEOR 4412 IEOR 4521 Mon 7pm 6 IEOR 4540 IEOR 4208 IDLE Wed 7pm 7 IEOR 4510 IDLE IDLE Tues 9am 8 IEOR 4707 IEOR 4102 IEOR 4574 Thurs 9am 9 IEOR 6712 IEOR 4601 IEOR 4507 Tues 1pm 10 IEOR 4211 IEOR 4523 IDLE Thurs 1pm 11 IEOR 4630 IEOR 4570 IDLE Tues 4pm 12 IEOR 4703 IEOR 3404 IEOR 4106 Thurs 4pm 13 IEOR 4720 IEOR 4405 IDLE Tues 7pm 14 IEOR 4524 IEOR 4578 IEOR 4577 Thurs 7pm 15 IEOR 4729 IEOR 4734 IDLE Fri 9am 16 IEOR 2261 IEOR 4525 IEOR 4998 Fri 1pm 17 IDLE IDLE IDLE Fri 4pm 18 IDLE IDLE IDLE Fri 7pm 19 IDLE IDLE IDLE
10 Case 2: Class Size Constraint For classes larger than 60 students, we determined that one room would not be large enough for the entire class. We broke these classes out into two exam slots, increasing our number of scheduled jobs to 55. This increases schedule Density to 92%; as the schedule density approaches 1, more jobs are completed late
11 Case 2 Schedule Number of Tardy Jobs: 15 Objective Function: 671 Percentage of Tardy Jobs: 27.3% Schedule Density: 91.7% Time Exam Slot Mudd 303 Mudd 545 Mudd 833 Mon 9 am 0 IEOR 4709a IEOR 4709b IEOR 3609a Wed 9am 1 IEOR 3402a IEOR 3609b IEOR 3402b Mon 1pm 2 IEOR 4307a IEOR 4307b IEOR Wed 1pm 3 IEOR 4418 IEOR 4004 IEOR 4573 Mon 4pm 4 IEOR 4404a IEOR 4700a IEOR 4404b Wed 4pm 5 IEOR 4700b IEOR IEOR 4208 Mon 7pm 6 IEOR 4540 IEOR 6614 IEOR 4412 Wed 7pm 7 IEOR 4510a IEOR 4510b IEOR 4521 Tues 9am 8 IEOR 4707a IEOR 4707b IEOR 4102a Thurs 9am 9 IEOR 4102b IEOR 4574 IEOR 6712 Tues 1pm 10 IEOR 4211 IEOR 4601 IEOR 4523 Thurs 1pm 11 IEOR 4630 IEOR 4507 IEOR 4570 Tues 4pm 12 IEOR 4703a IEOR 4703b IEOR 4106a Thurs 4pm 13 IEOR 3404a IEOR 4106b IEOR 3404b Tues 7pm 14 IEOR 4524 IEOR 4578 IEOR 4720 Thurs 7pm 15 IEOR 4729a IEOR 4729b IEOR 4405 Fri 9am 16 IEOR 2261a IEOR 2261b IEOR 4525a Fri 1pm 17 IEOR 4525b IEOR 4998 IEOR 4577 Fri 4pm 18 IEOR 4734 IDLE IDLE Fri 7pm 19 IDLE IDLE IDLE
12 Case 3: 4 rooms Case 3a: 4 rooms with a class size constraint Practical Benefits: Adding Mudd 633 to reduce schedule density See how adding additional classes affect a schedule with a lower schedule density Application: We ran WMDD and got a reasonable schedule that works as a good starting point to add more practical constraints
13 Case 3a Schedule Number of Tardy Jobs: 4 Objective Function: 129 Percentage of Tardy Jobs: 7.3% Schedule Density: 71.3% Time Exam Slot Mudd 303 Mudd 545 Mudd 833 Mudd 633 Mon 9 am 0 IEOR 4709a IEOR 4709b IEOR 3609a IEOR 3609b Wed 9am 1 IEOR 3402a IEOR 3402b IEOR 4418 IEOR 4573 Mon 1pm 2 IEOR 4307a IEOR 4307b IEOR IEOR 4004 Wed 1pm 3 IDLE IDLE IDLE IDLE Mon 4pm 4 IEOR 4404a IEOR 4700a IEOR 4404b IEOR 4700b Wed 4pm 5 IEOR IEOR 4208 IEOR 6614 IEOR 4412 Mon 7pm 6 IEOR 4540 IEOR 4521 IDLE IDLE Wed 7pm 7 IEOR 4510a IEOR 4510b IDLE IDLE Tues 9am 8 IEOR 4707a IEOR 4707b IEOR 4102a IEOR 4102b Thurs 9am 9 IEOR 4574 IEOR 6712 IEOR 4601 IEOR 4507 Tues 1pm 10 IEOR 4211 IEOR 4523 IDLE IDLE Thurs 1pm 11 IEOR 4630 IEOR 4570 IDLE IDLE Tues 4pm 12 IEOR 4703a IEOR 4703b IEOR 4106a IEOR 3404a Thurs 4pm 13 IEOR 4106b IEOR 3404b IEOR 4720 IEOR 4405 Tues 7pm 14 IEOR 4524 IEOR 4578 IEOR 4577 IEOR 4734 Thurs 7pm 15 IEOR 4729a IEOR 4729b IDLE IDLE Fri 9am 16 IEOR 2261a IEOR 2261b IEOR 4525a IEOR 4525b Fri 1pm 17 IEOR 4998 IDLE IDLE IDLE Fri 4pm 18 IDLE IDLE IDLE IDLE Fri 7pm 19 IDLE IDLE IDLE IDLE
14 Case 3: 4 rooms Case 3a: 4 rooms with a class size constraint Practical Benefits: Adding Mudd 633 to reduce schedule density See how adding additional classes affect a schedule with a lower schedule density Application: We ran WMDD and got a reasonable schedule that works as a good starting point to add more practical constraints Case 3b: Any split class still needs exams to happen at the same time. Practical Benefits: Prevent Academic Dishonesty Allows the professor to write only one exam Application: We ran WMDD, and then hardcoded a change for the 2 classes where this constraint was violated
15 Case 3b Schedule Number of Tardy Jobs: 4 Objective Function: 129 Percentage of Tardy Jobs: 7.3% Schedule Density: 71.3% Time Exam Slot Mudd 303 Mudd 545 Mudd 833 Mudd 633 Mon 9 am 0 IEOR 4709a IEOR 4709b IEOR 3609a IEOR 3609b Wed 9am 1 IEOR 3402a IEOR 3402b IEOR 4418 IEOR 4573 Mon 1pm 2 IEOR 4307a IEOR 4307b IEOR IEOR 4004 Wed 1pm 3 IDLE IDLE IDLE IDLE Mon 4pm 4 IEOR 4404a IEOR 4700a IEOR 4404b IEOR 4700b Wed 4pm 5 IEOR IEOR 4208 IEOR 6614 IEOR 4412 Mon 7pm 6 IEOR 4540 IEOR 4521 IDLE IDLE Wed 7pm 7 IEOR 4510a IEOR 4510b IDLE IDLE Tues 9am 8 IEOR 4707a IEOR 4707b IEOR 4102a IEOR 4102b Thurs 9am 9 IEOR 4574 IEOR 6712 IEOR 4601 IEOR 4507 Tues 1pm 10 IEOR 4211 IEOR 4523 IDLE IDLE Thurs 1pm 11 IEOR 4630 IEOR 4570 IDLE IDLE Tues 4pm 12 IEOR 4703a IEOR 4703b IEOR 4106a IEOR 4106b Thurs 4pm 13 IEOR 3404a IEOR 3404b IEOR 4720 IEOR 4405 Tues 7pm 14 IEOR 4524 IEOR 4578 IEOR 4577 IEOR 4734 Thurs 7pm 15 IEOR 4729a IEOR 4729b IDLE IDLE Fri 9am 16 IEOR 2261a IEOR 2261b IEOR 4525a IEOR 4525b Fri 1pm 17 IEOR 4998 IDLE IDLE IDLE Fri 4pm 18 IDLE IDLE IDLE IDLE Fri 7pm 19 IDLE IDLE IDLE IDLE
16 Case 4: 4 Rooms with Major track constraints Now we wanted to take into consideration students with a particular major track so that their exams do not overlap Simulation and Intro to Financial Engineering were scheduled at the same time, but juniors majoring in F.E. need to take both Production Planning and Advanced Optimization could have had the same final exam (both M/W 9am-12pm), and juniors need to take both Ran WMDD and then hardcoded a change for the 2 classes where this constraint was violated
17 Case 4 Schedule Number of Tardy Jobs: 4 Objective Function: 129 Percentage of Tardy Jobs: 7.3% Schedule Density: 71.3% Time Exam Slot Mudd 303 Mudd 545 Mudd 833 Mudd 633 Mon 9 am 0 IEOR 4709a IEOR 4709b IEOR 3609a IEOR 3609b Wed 9am 1 IEOR 3402a IEOR 3402b IEOR 4418 IEOR 4573 Mon 1pm 2 IEOR 4307a IEOR 4307b IEOR IEOR 4004 Wed 1pm 3 IDLE IDLE IDLE IDLE Mon 4pm 4 IEOR 4404a IEOR 6614 IEOR 4404b IEOR 4412 Wed 4pm 5 IEOR 4700a IEOR 4700b IEOR IEOR 4208 Mon 7pm 6 IEOR 4540 IEOR 4521 IDLE IDLE Wed 7pm 7 IEOR 4510a IEOR 4510b IDLE IDLE Tues 9am 8 IEOR 4707a IEOR 4707b IEOR 4102a IEOR 4102b Thurs 9am 9 IEOR 4574 IEOR 6712 IEOR 4601 IEOR 4507 Tues 1pm 10 IEOR 4211 IEOR 4523 IDLE IDLE Thurs 1pm 11 IEOR 4630 IEOR 4570 IDLE IDLE Tues 4pm 12 IEOR 4703a IEOR 4703b IEOR 4106a IEOR 3404a Thurs 4pm 13 IEOR 4106b IEOR 3404b IEOR 4720 IEOR 4405 Tues 7pm 14 IEOR 4524 IEOR 4578 IEOR 4577 IEOR 4734 Thurs 7pm 15 IEOR 4729a IEOR 4729b IDLE IDLE Fri 9am 16 IEOR 2261a IEOR 2261b IEOR 4525a IEOR 4525b Fri 1pm 17 IEOR 4998 IDLE IDLE IDLE Fri 4pm 18 IDLE IDLE IDLE IDLE Fri 7pm 19 IDLE IDLE IDLE IDLE
18 Final Conclusions Based on our results, we believe WMDD is a good starting point. Its simple application allows for a logical relaxation of the parallel problem and additional practical constraints can be added in on a case by case basis. Potential Issues: 1. As schedule density increases, the quality of schedule produced by WMDD does deteriorate 2. Larger jobs are placed early in the week, (ie on Monday rather than Wednesday), even though they don t need to be. We handle this issue in Case For exam scheduling, the number of late jobs might be a more accurate objective function than tardiness
19 Any Questions? Thanks for a great semester!
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