Scheduling - Expert Level: conflict resolution EXPERT

Step-by-step solution:

1. Total matches in double round-robin: 7 × (7-1) = 42
2. Maximum matches per round: 3
3. Minimum rounds: 42 ÷ 3 = 7 rounds

Answer: 7 rounds

Step-by-step solution (PERT):

1. Calculate expected duration for each task:
- Design: (4 + 4×6 + 8)/6 = 6.0
- Development: (3 + 4×4 + 7)/6 = 4.3
- Testing: (2 + 4×5 + 6)/6 = 4.7
- Deployment: (2 + 4×4 + 5)/6 = 3.8

2. Total expected duration: 18.8 days

Answer: 18.8 days

Step-by-step solution (Breakdown Recovery):

1. Original SPT order: Job A → Job B → Job D → Job E
2. Simulate processing with breakdown:
- Job A: 0 → 37
- Job B: 37 → 78
- Job D: Starts at 78, breakdown at 86 (8 min completed), repair 23 min, resume 49 min → completes at 158
- Job E: 158 → 233

3. Total makespan: 233 minutes
4. Delay caused by breakdown: 23 minutes

Answer: 233 minutes

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

Step-by-step solution:

Priority-Deadline Scheduling Algorithm:
1. Assign priority weights:
- High = 3, Medium = 2, Low = 1

2. Create priority-deadline table:
Task | Priority | Deadline | Duration
--------------|----------|----------|----------
Report | High | 5 | 3
Email | Low | 6 | 1
Presentation | High | 4 | 2
Analysis | Medium | 7 | 2

3. Sorting criteria:
- Primary: Highest priority first
- Secondary: Earliest deadline (if priority is same)

4. Sorted order:
1. Presentation (Priority: High, Deadline: 4)
2. Report (Priority: High, Deadline: 5)
3. Analysis (Priority: Medium, Deadline: 7)
4. Email (Priority: Low, Deadline: 6)

Answer: Presentation should be completed first

Key Strategy: Sort by priority first (descending), then by deadline (ascending) for tasks with equal priority.

Step-by-step solution:

1. Sort trains by arrival time
2. Greedy platform allocation
3. Maximum overlapping trains: 2

Answer: 2 platforms

Step-by-step solution:

1. Round Robin simulation:
2. Completion times:
- P1: 48
- P2: 51
- P4: 56
- P3: 61
- P5: 63

3. Average: 279 ÷ 5 = 55.8

Answer: 55.8

Step-by-step solution:

1. Total demand: 17
2. Vehicle capacity: 14
3. Minimum vehicles: ⌈17 ÷ 14⌉ = 2

Answer: 2 vehicles

Step-by-step solution:

1. Find intersection of availability:
Prof. C: [2, 4, 5, 8]
∩ Prof. A: [1, 7]
∩ Prof. E: [2, 3, 5, 8]
∩ Prof. D: [1, 2]
= ∅ (No common slots)

Answer: No common slot available

Step-by-step solution (Deductive Logic):

1. Apply Consecutive Constraint: 'C immediately after A' -> (A, C)
2. Apply Before Constraint: 'E before B'
3. Apply Exclusion Constraint: 'D not first or last'
4. Apply Gap Constraints: 'D is 3 after B'

Final Sequence: E → D → A → C → B

Answer: The task in the second position is D.

Key Strategy: Use fixed pairs and gap constraints to anchor positions.

Step-by-step solution (Redundancy Detection):

1. List all constraints:
1. Task 3 must be before Task 4
2. Task 4 must be before Task 1
3. Task 3 must be before Task 1
4. Task 2 must be after Task 1

2. Check for transitive relationships:
- From Constraint 1: Task 3 before Task 4
- From Constraint 2: Task 4 before Task 1
- By transitivity: Task 3 before Task 1
- This makes Constraint 3 unnecessary (redundant)

3. Verify other constraints are independent:
- Constraint 4 (Task 2 after Task 1) adds unique information

Answer: Constraint 3 is redundant

Key Strategy: Look for transitive relationships (A→B, B→C implies A→C).

Step-by-step solution:

Table Method with Constraint Elimination:
1. Create a 3D table: Days x Shifts x Employees

2. Apply direct constraints:
- Monday Morning: Alice (fixed)
- Tuesday Evening: Charlie (fixed)
- Bob: Never Night shift (all days)

3. Apply rotation constraint:
- Alice (Morning Mon) cannot be Morning Tue
- Charlie (Evening Tue) cannot be Evening Wed

4. Fill Monday:
- Morning: Alice
- Evening: Charlie (can work evening)
- Night: Diana (Bob can't do night)

5. Fill Tuesday:
- Morning: Bob (Alice can't repeat, Charlie is evening)
- Evening: Charlie (fixed)
- Night: Diana (Bob can't)

6. Fill Wednesday:
- Charlie can't be Evening (was Evening Tue)
- Alice can be Evening (was Morning Mon, okay to shift)
- Answer: Alice works Evening on Wednesday

Key Strategy: Apply fixed constraints first, then use rotation rules to eliminate impossible assignments systematically.

Step-by-step solution:

1. Identify peak demand: 9 staff at hour 3
2. Staff can work multiple hours → schedule around peak
3. Minimum staff needed: 9

Answer: 9 staff

Step-by-step solution (JIT Penalty Calculation):

1. EDD Sequence: Component B → Component D → Component A → Component C
2. Calculate completion times and penalties:
- Component B: completes at 40, due 78, early by 38 min → penalty 38
- Component D: completes at 76, due 92, early by 16 min → penalty 16
- Component A: completes at 111, due 99, late by 12 min → penalty 192
- Component C: completes at 151, due 120, late by 31 min → penalty 527

3. Total penalty: 773

Answer: 773 penalty points

Key Strategy: JIT scheduling minimizes total earliness + tardiness penalties, balancing inventory costs and customer satisfaction.

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:

1. **Model as edge coloring of complete graph K_6
2. Vizing's theorem: χ'(K_n) = n-1 for even n, n for odd n
3. For 6 teams: 5 colors/rounds needed

Answer: 5 rounds

Step-by-step solution:

Timeline Planning Method:
1. Calculate total preparation time needed:
- Physics: 2 days
- Mathematics: 3 days
- Chemistry: 2 days
- Total: 7 days

2. Available days: 10 days
3. Extra buffer days: 3 days
4. Optimal schedule:
- Days 1-2: Prepare for Physics
- Day 3: Physics exam
- Days 4-6: Prepare for Mathematics
- Day 7: Mathematics exam
- Days 8-9: Prepare for Chemistry
- Day 10: Chemistry exam

Answer: The last exam will be on Day 10

Key Strategy: Schedule exams immediately after preparation period ends, accounting for all required prep days.

Step-by-step solution:

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

2. Courses without prerequisites (can take first): CS401

Answer: CS401

Step-by-step solution:

1. Single elimination principle: Each match eliminates exactly one team
2. Teams to eliminate: 8 - 1 = 7 teams must be eliminated
3. Matches needed: 7 matches

Answer: 7 matches

Step-by-step solution:

1. Patients: 11
2. Capacity per nurse: 4
3. Minimum nurses: ⌈11 ÷ 4⌉ = 3

Answer: 3 nurses

Step-by-step solution:

Network Path Analysis:
1. Identify all possible routes from City A to City D:
- City A→City B -> City B→City D
- City A→City C -> City C→City D
- City A→City B -> City B→City C -> City C→City D

2. Best route found:
- T1: City A to City B (09:30 - 1:09 PM)
- Connection time: 51 minutes
- T3: City B to City D (14:00 - 4:37 PM)

Earliest arrival: 4:37 PM

Key Strategy: Enumerate all possible routes, verify connection times meet minimum requirements.