Scheduling - Beginner-Intermediate Level: schedule constraints BEGINNER-INTERMEDIATE

Step-by-step solution:

Set Intersection Method:
1. List all availability:
- Alex: {9:00 AM, 10:00 AM, 2:00 PM, 3:00 PM}
- Ben: {10:00 AM, 11:00 AM, 2:00 PM}
- Cara: {9:00 AM, 11:00 AM, 12:00 PM, 3:00 PM}
- Diana: {10:00 AM, 12:00 PM, 2:00 PM, 3:00 PM}

2. Find common slots (intersection):
- Common to all = Alex AND Ben AND Cara AND Diana
- Result: Empty set (No common time)

3. Conclusion: No common time slot available

Key Strategy: Use set intersection to find common availability, then choose the earliest time.

Step-by-step solution:

Timeline Grid Method:
1. Create month-date grid:
Jan 5 | Jan 15 | Mar 5 | Mar 15 | May 5 | May 15 | Jul 5 | Jul 15

2. Apply constraints:
- T in March (Mar 5 or Mar 15)
- P on 15th (any month, date 15)
- Two people between S and V
(If S at position 1, V at position 4)
- R and W in same month

3. Systematic placement:
- Place T at Mar 5 (satisfies March constraint)
- Place P at Mar 15 (satisfies 15th constraint)
- For 'two between' constraint: If S at Jan 5, V at Mar 15
- R and W together: May 5 & May 15

4. Verification:
All constraints satisfied with S in March

Key Strategy: Use grid to visualize all slots, apply direct constraints first, then deduce positions using gap constraints.

Step-by-step solution:

1. Identify courses with prerequisites:
- CS101 needs CS201
- CS301 needs CS201
- CS102 needs CS201
- CS401 needs CS101

2. Courses without prerequisites (can take first): CS201

Answer: CS201

Step-by-step solution:

1. Patients: 12
2. Capacity per nurse: 4
3. Minimum nurses: ⌈12 ÷ 4⌉ = 3

Answer: 3 nurses

Step-by-step solution:

1. Identify courses with prerequisites:
- CS401 needs CS301
- CS202 needs CS301
- CS102 needs CS401
- CS201 needs CS401

2. Courses without prerequisites (can take first): CS301

Answer: CS301

Step-by-step solution:

1. Total invigilator slots per time: 2
2. Minimum teachers needed: At least 2 (one per invigilator slot)
3. Same teachers can invigilate multiple slots

Answer: 2 teachers

Step-by-step solution:

1. Convert all times to minutes for easier calculation:
- Flight 101: Departs at 8:00, Arrives at 10:00
- Flight 102: Departs at 10:30, Arrives at 12:30
- Flight 103: Departs at 13:00, Arrives at 15:00
- Flight 104: Departs at 15:30, Arrives at 17:30
- Flight 105: Departs at 18:00, Arrives at 20:00

2. Duty time limit: 480 minutes (8 hours)
3. Minimum connection time: 30 minutes

4. Find optimal sequence of flights:
Best sequence found: Flight 103 → Flight 104 → Flight 105
- Take Flight 103: Departs at 13:00
- Connection time: 30 minutes
- Take Flight 104: Departs at 15:30
- Connection time: 30 minutes
- Take Flight 105: Departs at 18:00

Total duty time: 420 minutes (7 hours, 0 minutes)

5. Maximum flights possible: 3

Answer: 3 flights

✓ Duty time check: 7h 0m ≤ 8h (PASSED)

Step-by-step solution:

1. Round Robin simulation:
2. Completion times:
- P1: 32
- P2: 33
- P3: 37

3. Average: 102 ÷ 3 = 34.0

Answer: 34.0

Step-by-step solution (Interval Graph):

1. Plot intervals on timeline:
Event A: ████ from 4 to 8
Event B: ███████ from 6 to 13
Event C: ███ from 9 to 12
Event D: ███ from 13 to 16
Event E: ██ from 20 to 22
Event F: ███ from 1 to 4

2. Find maximum overlap:
Maximum 2 events overlap at once

Answer: 2 rooms needed

Step-by-step solution:

1. Find intersection of availability:
Prof. A: [2, 5, 6]
∩ Prof. E: [2, 4, 5, 6]
∩ Prof. C: [1, 6, 8]
∩ Prof. D: [4, 5, 8]
= ∅ (No common slots)

Answer: No common slot available

Step-by-step solution:

1. Total invigilator slots per time: 2
2. Minimum teachers needed: At least 2 (one per invigilator slot)
3. Same teachers can invigilate multiple slots

Answer: 2 teachers

Data Sufficiency Reasoning:

Step 1 - Analyze Statement (1) alone: Physics is on Wednesday, two days after Chemistry.
This gives partial information but not enough to determine the answer uniquely.

Step 2 - Analyze Statement (2) alone: Mathematics is on Friday, immediately after Biology.
This also gives partial information insufficient by itself.

Step 3 - Combine statements:
Together, they provide enough constraints to solve uniquely.

Conclusion: Either statement alone is sufficient.

Key Strategy: Test each statement independently first, then combine only if neither alone works.

Step-by-step solution (Breakdown Recovery):

1. Original SPT order: Job D → Job E → Job C → Job A
2. Simulate processing with breakdown:
- Job D: 0 → 36
- Job E: Starts at 36, breakdown at 79 (43 min completed), repair 23 min, resume 10 min → completes at 112
- Job C: 112 → 173
- Job A: 173 → 259

3. Total makespan: 259 minutes
4. Delay caused by breakdown: 23 minutes

Answer: 259 minutes

Key Strategy: Simulate the timeline, account for breakdown during active job processing.

Step-by-step solution:

1. Round-robin schedule using circle method
2. Round 2 matches:
- Team C vs Team E
- Team A vs Team D
- Team F vs Team B

3. Team C plays against Team E

Answer: Team E

Step-by-step solution:

1. Patients: 15
2. Capacity per nurse: 4
3. Minimum nurses: ⌈15 ÷ 4⌉ = 4

Answer: 4 nurses

Step-by-step solution:

1. Total on-call days: 30
2. Base days per doctor: 30 ÷ 4 = 7 days
3. Remainder: 2 doctor(s) get one extra day

Answer: 7 days, with 2 doctor(s) getting 8 days

Step-by-step solution (Learning Curve):

1. Learning curve formula: T_n = T_1 × n^-0.152
where exponent = log(0.9)/log(2) = -0.152

2. Calculate cumulative time using integration:
Cumulative time for N units = T_1 × N^0.84799690655495 / (learning_exponent + 1)

3. Time per batch:
Batch 1 (20 units): 69.6 minutes
Batch 2 (30 units): 54.5 minutes
Batch 3 (10 units): 50.6 minutes

4. Total time: 174.6 ≈ 175 minutes

Key Strategy: Learning curve reduces time with repetition; use cumulative average method for batch calculations.

Step-by-step solution:

1. Identify courses with prerequisites:
- CS201 needs CS301
- CS202 needs CS201
- CS401 needs CS202
- CS101 needs CS202

2. Courses without prerequisites (can take first): CS301

Answer: CS301

Step-by-step solution:

1. Single round-robin formula: n(n-1)/2
2. For 6 teams: 6 × 5 / 2 = 15 matches
3. Double round-robin: 15 × 2 = 30 matches

Answer: 30 matches

Step-by-step solution:

1. Rotation cycle: 5 employees × 5 days = 25 days
2. Verification: Each employee cycles through all shifts

Answer: 25 days