Scheduling - Beginner Level: appointment logic BEGINNER

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

1. EDD Sequence: Component C → Component B → Component A → Component D
2. Calculate completion times and penalties:
- Component C: completes at 31, due 49, early by 18 min → penalty 90
- Component B: completes at 53, due 70, early by 17 min → penalty 34
- Component A: completes at 107, due 90, late by 17 min → penalty 187
- Component D: completes at 163, due 116, late by 47 min → penalty 705

3. Total penalty: 1016

Answer: 1016 penalty points

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

Step-by-step solution (Graph Coloring):

1. Model as graph coloring problem:
- Vertices = Courses
- Edges = Conflicts (courses that cannot be together)
2. Apply greedy coloring algorithm:
- Chemistry: Slot 1
- Math: Slot 2
- CS: Slot 3
- Physics: Slot 4
- History: Slot 3
- English: Slot 1

3. Colors/slots used: 4

Answer: Minimum 4 time slots

Key Strategy: The chromatic number of the conflict graph gives the minimum slots needed.

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. Patients: 9
2. Capacity per nurse: 5
3. Minimum nurses: ⌈9 ÷ 5⌉ = 2

Answer: 2 nurses

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:

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.

Data Sufficiency Reasoning:

Step 1 - Analyze Statement (1) alone: Testing starts exactly 5 days after Design ends.
This gives partial information but not enough to determine the answer uniquely.

Step 2 - Analyze Statement (2) alone: Planning takes 3 days and ends before Design starts.
This also gives partial information insufficient by itself.

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

Conclusion: Both statements together are still insufficient.

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

Step-by-step solution:

1. Total machines: 4
2. Maintenance duration: 2 day
3. Staggered schedule: Never have all machines down simultaneously
4. Maximum operational: 4 - 1 = 3

Answer: 3 machines

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:

Timeline Approach:
1. Convert all times to 24-hour format for easy comparison
- Dinner: 8:00 PM
- Morning Yoga: 6:00 AM
- Lunch Break: 1:00 PM
- Office Work: 9:00 AM

2. Arrange in chronological order:
1. Lunch Break at 1:00 PM
2. Morning Yoga at 6:00 AM
3. Dinner at 8:00 PM
4. Office Work at 9:00 AM

Final Schedule: Lunch Break -> Morning Yoga -> Dinner -> Office Work

Key Strategy: Convert all times to 24-hour format and arrange from earliest to latest.

Step-by-step solution:

Conflict Detection Method:
1. Create doctor-time matrix:
Doctor | 10:00 | 10:30 | 11:00 | 11:30
------------|-------|-------|-------|-------
Dr. Patel | A | C | E | ---
Dr. Kumar | B | --- | --- | ---
Dr. Shah | D | --- | --- | ---

2. Check for conflicts:
- Dr. Patel at 10:00: Only Patient A (No conflict)
- Dr. Patel at 10:30: Only Patient C (No conflict)
- Dr. Patel at 11:00: Only Patient E (No conflict)
- Dr. Kumar at 10:00: Only Patient B (No conflict)
- Dr. Shah at 10:00: Only Patient D (No conflict)

3. Verification:
- No doctor has multiple patients in same slot
- All preferences can be honored

Answer: 0 patients need rescheduling

Key Strategy: Map all appointments to a doctor-time grid and identify slots where multiple patients request the same doctor.

Step-by-step solution:

1. Total surgery time: 405 min = 6.8 hours
2. Available OR hours: 2 × 8 = 16 hours
3. Total ≤ Available → Can complete in one day

Answer: All surgeries can be scheduled within one day

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

1. EDD Sequence: Component A → Component B → Component C → Component D
2. Calculate completion times and penalties:
- Component A: completes at 43, due 64, early by 21 min → penalty 42
- Component B: completes at 90, due 64, late by 26 min → penalty 286
- Component C: completes at 148, due 82, late by 66 min → penalty 1188
- Component D: completes at 185, due 116, late by 69 min → penalty 1173

3. Total penalty: 2689

Answer: 2689 penalty points

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

Step-by-step solution (Breakdown Recovery):

1. Original SPT order: Job B → Job A → Job D → Job C
2. Simulate processing with breakdown:
- Job B: 0 → 70
- Job A: Starts at 70, breakdown at 90 (20 min completed), repair 22 min, resume 65 min → completes at 177
- Job D: 177 → 264
- Job C: 264 → 353

3. Total makespan: 353 minutes
4. Delay caused by breakdown: 22 minutes

Answer: 353 minutes

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

Step-by-step solution (Breakdown Recovery):

1. Original SPT order: Job E → Job B → Job D → Job A
2. Simulate processing with breakdown:
- Job E: 0 → 48
- Job B: Starts at 48, breakdown at 81 (33 min completed), repair 25 min, resume 27 min → completes at 133
- Job D: 133 → 199
- Job A: 199 → 272

3. Total makespan: 272 minutes
4. Delay caused by breakdown: 25 minutes

Answer: 272 minutes

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

Step-by-step solution:

1. Rotation cycle: 6 employees × 7 days = 42 days
2. Verification: Each employee cycles through all shifts

Answer: 42 days

Step-by-step solution:

1. Round Robin simulation:
2. Completion times:
- P3: 24
- P1: 30
- P2: 33

3. Average: 87 ÷ 3 = 29.0

Answer: 29.0

Step-by-step solution:

Timeline Approach:
1. Convert all times to 24-hour format for easy comparison
- Breakfast: 7:30 AM
- Lunch Break: 1:00 PM
- Dinner: 8:00 PM
- Morning Yoga: 6:00 AM

2. Arrange in chronological order:
1. Lunch Break at 1:00 PM
2. Morning Yoga at 6:00 AM
3. Breakfast at 7:30 AM
4. Dinner at 8:00 PM

Final Schedule: Lunch Break -> Morning Yoga -> Breakfast -> Dinner

Key Strategy: Convert all times to 24-hour format and arrange from earliest to latest.

Step-by-step solution (PERT):

1. Calculate expected duration for each task:
- Design: (3 + 4×6 + 7)/6 = 5.7
- Development: (3 + 4×6 + 8)/6 = 5.8
- Testing: (2 + 4×4 + 6)/6 = 4.0
- Deployment: (4 + 4×7 + 9)/6 = 6.8

2. Total expected duration: 22.3 days

Answer: 22.3 days

Step-by-step solution:

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

Answer: 2 platforms