Scheduling Worksheet 3 - Beginner Level (day scheduling) | 20 Questions

Question 1

A football league has 5 teams. Each team plays every other team twice (home and away). What is the minimum number of rounds needed if each round has the maximum possible matches?

Step-by-step solution:

1. Total matches in double round-robin: 5 × (5-1) = 20
2. Maximum matches per round: 2
3. Minimum rounds: 20 ÷ 2 = 5 rounds

Answer: 5 rounds

league, fixture, home-away, sports, optimization

Question 2

A machine must process four products (P1, P2, P3, P4) to minimize total time. Processing times: P1: 57, P2: 66, P3: 37, P4: 68 Setup times: | From ↓ / To → | P1 | P2 | P3 | P4 | | P1 | 0 | 27 | 24 | 22 | | P2 | 23 | 0 | 26 | 38 | | P3 | 23 | 15 | 0 | 15 | | P4 | 17 | 20 | 20 | 0 | What sequence minimizes total time?

Optimal sequence examined for all 24 permutations.
Minimum time: 284 minutes. Sequence: P4 - P1 - P3 - P2.
Calculation: P4 → P1 → P3 → P2.
Time: PT(P4) + (ST(P4→P1) + PT(P1)) + ...

professional, sequencing, optimization, TSP-lite

Question 3

A manager has 4 tasks to complete over 8 working hours. The task details are: - Report: Priority High, Duration 3 hours, Deadline 5 hours - Email: Priority Low, Duration 1 hours, Deadline 6 hours - Presentation: Priority High, Duration 2 hours, Deadline 4 hours - Analysis: Priority Medium, Duration 2 hours, Deadline 7 hours If tasks are scheduled based on priority first and deadline second, which task should be completed first?

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.

priority-based, deadline-driven, task-ordering

Question 4

Given these scheduling constraints: - Task C must be before Task B - Task A must be after Task B - Task D must be immediately after Task C Is a valid schedule possible?

Step-by-step solution:

1. Check for cycles: No circular dependencies
2. Check immediate constraints: Can be satisfied
3. Conclusion: Yes, a valid schedule exists

Answer: Yes, a valid schedule exists

feasibility, satisfiability, constraint-check, logic, verification

Question 5

Hospital OR scheduling with 2 operating rooms (8 hours each): - Emergency: 40 min, Priority 1 - Urgent: 73 min, Priority 2 - Elective A: 81 min, Priority 3 - Elective B: 64 min, Priority 3 - Routine: 109 min, Priority 4 Can all surgeries be completed in one day?

Step-by-step solution:

1. Total surgery time: 367 min = 6.1 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

surgery, operating-room, hospital, healthcare, critical

Question 6

Project scheduling with two objectives: minimize time and minimize cost. - Schedule A: 100 days, $50K - Schedule B: 120 days, $40K - Schedule C: 90 days, $60K - Schedule D: 110 days, $45K - Schedule E: 95 days, $55K Which schedules are on the Pareto frontier (not dominated in both objectives)?

Step-by-step solution:

1. Pareto dominance: Schedule X dominates Y if X is better in at least one objective and not worse in others
2. Pareto frontier schedules: Schedule A, Schedule B, Schedule C, Schedule D, Schedule E

Answer: Schedule A, Schedule B, Schedule C, Schedule D, Schedule E

pareto, multi-objective, trade-off, advanced, optimization

Question 7

A school needs to schedule 6 courses. The following courses have overlapping students and cannot be scheduled at the same time: - English conflicts with CS - English conflicts with Biology - English conflicts with Chemistry - English conflicts with Physics - CS conflicts with Biology - CS conflicts with Chemistry - CS conflicts with Physics - Chemistry conflicts with Physics What is the minimum number of time slots needed to schedule all courses without conflicts?

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:
- History: Slot 1
- English: Slot 1
- CS: Slot 2
- Chemistry: Slot 3
- Biology: Slot 3
- Physics: Slot 4

3. Colors/slots used: 4

Answer: Minimum 4 time slots

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

timetable, conflict-resolution, graph-theory, chromatic-number, advanced

Question 8

A company has 8 employees working in 3 shifts. Shifts rotate every 7 days. After how many days does an employee return to the same shift pattern?

Step-by-step solution:

1. Rotation cycle: 8 employees × 7 days = 56 days
2. Verification: Each employee cycles through all shifts

Answer: 56 days

rotation, cyclic, shift-pattern, workforce, healthcare

Question 9

A job shop has 3 machines. Jobs and their routes: - Job A: M1 → M3 → M2 with times 35, 10, 20 - Job B: M2 → M1 → M3 with times 19, 14, 30 - Job C: M1 → M3 → M2 with times 28, 30, 26 What is a lower bound on the minimum makespan?

Step-by-step solution:

1. Machine load bound: 77
2. Job processing bound: 84
3. Lower bound: 84

Answer: 84

job-shop, routing, complex, np-hard, advanced

Question 10

Project tasks with uncertain durations (optimistic, likely, pessimistic) in days: - Design: (2, 5, 6) - Development: (3, 4, 7) - Testing: (2, 5, 7) - Deployment: (3, 6, 8) Using the PERT formula (O + 4M + P)/6, what is the expected total project duration?

Step-by-step solution (PERT):

1. Calculate expected duration for each task:
- Design: (2 + 4×5 + 6)/6 = 4.7
- Development: (3 + 4×4 + 7)/6 = 4.3
- Testing: (2 + 4×5 + 7)/6 = 4.8
- Deployment: (3 + 4×6 + 8)/6 = 5.8

2. Total expected duration: 19.7 days

Answer: 19.7 days

fuzzy, uncertainty, duration, advanced, real-world

Question 11

6 tasks (A, B, C, D, E, F) are to be completed one after the other. The following conditions must be met: - Task F must be performed immediately after Task A. - Task D must be completed before Task E. - Task C is neither the first nor the last task to be completed. - Task E is performed exactly 2 positions after Task B. Which task is scheduled in the fifth position?

Step-by-step solution (Deductive Logic):

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

Final Sequence: A → F → C → B → D → E

Answer: The task in the fifth position is D.

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

linear-arrangement, relative-constraints, deduction, CAT-level

Question 12

A project consists of the following tasks: - Task A (Requirements Analysis): 2 days, Depends on: None - Task B (Design): 3 days, Depends on: A - Task C (Database Setup): 2 days, Depends on: A - Task D (Development): 5 days, Depends on: B, C - Task E (Testing): 3 days, Depends on: D What is the minimum number of days required to complete the entire project?

Step-by-step solution:

Critical Path Method (CPM):
1. Identify dependencies and calculate earliest start times:
- Task A: Starts on Day 0, Duration 2 days
Finishes on Day 2
- Task B: Starts on Day 2, Duration 3 days
Finishes on Day 5
- Task C: Starts on Day 2, Duration 2 days
Finishes on Day 4
- Task D: Starts on Day 5, Duration 5 days
Finishes on Day 10
- Task E: Starts on Day 10, Duration 3 days
Finishes on Day 13

2. Task timeline:
Task A: Days 0-2
Task B: Days 2-5 (after A)
Task C: Days 2-4 (after A)
Task D: Days 5-10 (after B and C)
Task E: Days 10-13 (after D)

3. Critical path: A -> B -> D -> E (or A -> C -> D -> E)
4. Total project duration: 13 days

Key Strategy: Calculate earliest start time for each task based on predecessor completion times; the longest path determines total duration.

project-management, dependencies, critical-path, professional

Question 13

A company has 5 employees working in 3 shifts. Shifts rotate every 10 days. After how many days does an employee return to the same shift pattern?

Step-by-step solution:

1. Rotation cycle: 5 employees × 10 days = 50 days
2. Verification: Each employee cycles through all shifts

Answer: 50 days

rotation, cyclic, shift-pattern, workforce, healthcare

Question 14

A bus route takes 53 minutes one-way. Peak frequency: 1 bus every 12 minutes. What is the minimum number of buses needed to maintain this frequency in both directions?

Step-by-step solution:

1. Round trip time: 53 × 2 = 106 minutes
2. Headway: 12 minutes
3. Buses needed: ⌈106 ÷ 12⌉ = 9

Answer: 9 buses

bus, timetable, frequency, public-transport, optimization

Question 15

A project consists of the following tasks: - Task A (Requirements Analysis): 2 days, Depends on: None - Task B (Design): 3 days, Depends on: A - Task C (Database Setup): 2 days, Depends on: A - Task D (Development): 5 days, Depends on: B, C - Task E (Testing): 3 days, Depends on: D What is the minimum number of days required to complete the entire project?

Step-by-step solution:

Critical Path Method (CPM):
1. Identify dependencies and calculate earliest start times:
- Task A: Starts on Day 0, Duration 2 days
Finishes on Day 2
- Task B: Starts on Day 2, Duration 3 days
Finishes on Day 5
- Task C: Starts on Day 2, Duration 2 days
Finishes on Day 4
- Task D: Starts on Day 5, Duration 5 days
Finishes on Day 10
- Task E: Starts on Day 10, Duration 3 days
Finishes on Day 13

2. Task timeline:
Task A: Days 0-2
Task B: Days 2-5 (after A)
Task C: Days 2-4 (after A)
Task D: Days 5-10 (after B and C)
Task E: Days 10-13 (after D)

3. Critical path: A -> B -> D -> E (or A -> C -> D -> E)
4. Total project duration: 13 days

Key Strategy: Calculate earliest start time for each task based on predecessor completion times; the longest path determines total duration.

project-management, dependencies, critical-path, professional

Question 16

A manager has 4 tasks to complete over 8 working hours. The task details are: - Report: Priority High, Duration 3 hours, Deadline 5 hours - Email: Priority Low, Duration 1 hours, Deadline 6 hours - Presentation: Priority High, Duration 2 hours, Deadline 4 hours - Analysis: Priority Medium, Duration 2 hours, Deadline 7 hours If tasks are scheduled based on priority first and deadline second, which task should be completed first?

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.

priority-based, deadline-driven, task-ordering

Question 17

A football league has 6 teams. Each team plays every other team twice (home and away). What is the minimum number of rounds needed if each round has the maximum possible matches?

Step-by-step solution:

1. Total matches in double round-robin: 6 × (6-1) = 30
2. Maximum matches per round: 3
3. Minimum rounds: 30 ÷ 3 = 5 rounds

Answer: 5 rounds

league, fixture, home-away, sports, optimization

Question 18

A machine needs to process 4 jobs. Processing times: - Job D: 72 minutes - Job C: 74 minutes - Job E: 79 minutes - Job B: 76 minutes The machine breaks down at 103 minutes and takes 20 minutes to repair. Jobs are scheduled using Shortest Processing Time (SPT) first rule. What is the total completion time (makespan) after handling the breakdown?

Step-by-step solution (Breakdown Recovery):

1. Original SPT order: Job D → Job C → Job B → Job E
2. Simulate processing with breakdown:
- Job D: 0 → 72
- Job C: Starts at 72, breakdown at 103 (31 min completed), repair 20 min, resume 43 min → completes at 166
- Job B: 166 → 242
- Job E: 242 → 321

3. Total makespan: 321 minutes
4. Delay caused by breakdown: 20 minutes

Answer: 321 minutes

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

manufacturing, breakdown, recovery, simulation, reliability

Question 19

In a double round-robin tournament with 6 teams, each team plays every other team twice (once home, once away). How many home matches does each team play?

Step-by-step solution:

1. Total matches per team in double round-robin: 2 × (6 - 1) = 10 matches
2. Half are home matches: 10 ÷ 2 = 5 home matches

Answer: 5 home matches

tournament, sports, combinations, fairness, competitive-exams

Question 20

A school has 5 exams in 3 time slots. Each time slot needs 2 invigilators. A teacher can invigilate at most one exam per time slot. What is the minimum number of teachers required?

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

exam, invigilation, proctor, academic, constraints

Scheduling - Beginner Level: day scheduling BEGINNER

What you'll learn in this worksheet: