Scheduling Worksheet 17 - Intermediate Level (time allocation) | 20 Questions

Question 1

A hospital needs one doctor on-call each day for 30 days. There are 5 doctors: Dr. Brown, Dr. Jones, Dr. Lee, Dr. Patel, Dr. Smith. If the schedule is as fair as possible, how many days will each doctor be on-call?

Step-by-step solution:

1. Total on-call days: 30
2. Base days per doctor: 30 ÷ 5 = 6 days
3. Remainder: 0 doctor(s) get one extra day

Answer: 6 days each

on-call, availability, emergency, medical, fairness

Question 2

A project involves two events, Event A (Meeting) and Event B (Training). The constraints are: - **Event A:** Duration 90 minutes. Must start between 9:00 AM and 11:00 AM. - **Event B:** Duration 60 minutes. Must finish by 3:00 PM. - **Gap:** A minimum of 2 hours is required between the end of Event A and the start of Event B. Assuming all constraints must be met, what is the earliest possible start time for Event B?

Step-by-step solution (Time Arithmetic):

1. Goal: To find the earliest start time for Event B, we must use the earliest possible schedule for Event A.
2. Calculate Earliest Finish Time for Event A:
- Earliest Start for A: 9:00 AM
- Duration of A: 90 minutes (1 hour 30 minutes)
- Earliest Finish for A: 9:00 AM + 1 hour 30 minutes = 10:30 AM.
3. Apply Minimum Gap:
- Earliest Start for B = (Earliest Finish A) + (Minimum Gap)
- Minimum Gap: 2 hours (120 minutes)
- Earliest Start for B: 10:30 AM + 2 hours = 12:30 PM.
4. Check Deadline for Event B:
- If B starts at 12:30 PM, its finish time is 12:30 PM + 60 minutes = 1:30 PM.
- The latest finish time for B is 3:00 PM. Since 1:30 PM is before 3:00 PM, the schedule is valid.
Answer: The earliest possible start time for Event B is 12:30 PM.
Key Strategy: To find the minimum time for the second event, use the minimum time for the first event, plus the mandatory gap.

time-arithmetic, gap-constraint, optimization, banking-exams

Question 3

A factory has 3 production lines: Line 1, Line 2, Line 3. Three products require the following operations: **Product X:** - Cut: 30 min on Line 2 - Assemble: 45 min on Line 1 - Package: 15 min on Line 2 **Product Y:** - Cut: 20 min on Line 1 - Assemble: 60 min on Line 3 - Package: 20 min on Line 1 **Product Z:** - Cut: 40 min on Line 2 - Assemble: 30 min on Line 3 - Package: 25 min on Line 3 All products must be completed (all 3 operations each). Multiple operations can run in parallel on different lines. Which production line is the bottleneck, and what is its total load (in minutes)?

Step-by-step solution (Bottleneck Analysis):

1. Calculate total load per production line:
- Line 1: 85 minutes
- Line 2: 85 minutes
- Line 3: 115 minutes

2. Identify bottleneck: The line with maximum load = Line 3
3. Bottleneck load: 115 minutes

Answer: Line 3 (115 minutes)

Key Strategy: The bottleneck determines maximum throughput; optimize the bottleneck first for overall efficiency.

nested, bottleneck, resource-allocation, multi-line, factory

Question 4

Four team members (Eva, Frank, Charlie, Bob) must be assigned to four unique tasks (Testing, Deployment, Design, Documentation). The assignments must follow these rules: 1. Eva must handle Testing. 2. Frank cannot handle Documentation. 3. Charlie and Bob must be adjacent in (Testing → Deployment → Design → Documentation). Based on the constraints, which statement MUST be true?

No valid schedule found given the constraints. The only guaranteed assignment is: Eva must handle Testing.
If the constraints cannot all be satisfied, fallback is to force rule 1's assignment.

multi-person, shift-scheduling, constraint-satisfaction, puzzle

Question 5

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 6

In a round-robin tournament with 6 teams, in Round 4, Team B plays against which team?

Step-by-step solution:

1. Round-robin schedule using circle method
2. Round 4 matches:
- Team C vs Team E
- Team B vs Team F
- Team A vs Team D

3. Team B plays against Team F

Answer: Team F

tournament, sports, combinations, fairness, competitive-exams

Question 7

A PhD thesis defense requires all 3 committee members to be present. Their availability (slots 1-8): - Prof. D: Slots 3, 7, 4 - Prof. C: Slots 5, 1, 7, 8 - Prof. E: Slots 8, 2, 6 What is the earliest slot when all can attend?

Step-by-step solution:

1. Find intersection of availability:
Prof. D: [3, 4, 7]
∩ Prof. C: [1, 5, 7, 8]
∩ Prof. E: [2, 6, 8]
= ∅ (No common slots)

Answer: No common slot available

thesis, defense, committee, academic, availability

Question 8

Eight people attend seminars in four different months (January, March, May, July) on two dates (5th and 15th). Two people attend per month. The constraints are: - W attends in March - U attends on the 15th - Exactly two people attend between P and T - S attends in the same month as V In which month does P attend?

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

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

4. Verification:
All constraints satisfied with P in March

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

month-based, date-based, gap-constraints, banking-exams

Question 9

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 10

A PhD thesis defense requires all 4 committee members to be present. Their availability (slots 1-8): - Prof. A: Slots 2, 8, 6, 5 - Prof. E: Slots 4, 8, 3, 1 - Prof. C: Slots 3, 8, 1, 7 - Prof. B: Slots 1, 3, 6 What is the earliest slot when all can attend?

Step-by-step solution:

1. Find intersection of availability:
Prof. A: [2, 5, 6, 8]
∩ Prof. E: [1, 3, 4, 8]
∩ Prof. C: [1, 3, 7, 8]
∩ Prof. B: [1, 3, 6]
= ∅ (No common slots)

Answer: No common slot available

thesis, defense, committee, academic, availability

Question 11

A factory has 3 production lines: Line 1, Line 2, Line 3. Three products require the following operations: **Product X:** - Cut: 30 min on Line 3 - Assemble: 45 min on Line 3 - Package: 15 min on Line 3 **Product Y:** - Cut: 20 min on Line 1 - Assemble: 60 min on Line 2 - Package: 20 min on Line 3 **Product Z:** - Cut: 40 min on Line 3 - Assemble: 30 min on Line 3 - Package: 25 min on Line 1 All products must be completed (all 3 operations each). Multiple operations can run in parallel on different lines. Which production line is the bottleneck, and what is its total load (in minutes)?

Step-by-step solution (Bottleneck Analysis):

1. Calculate total load per production line:
- Line 1: 45 minutes
- Line 2: 60 minutes
- Line 3: 180 minutes

2. Identify bottleneck: The line with maximum load = Line 3
3. Bottleneck load: 180 minutes

Answer: Line 3 (180 minutes)

Key Strategy: The bottleneck determines maximum throughput; optimize the bottleneck first for overall efficiency.

nested, bottleneck, resource-allocation, multi-line, factory

Question 12

A project involves two events, Event A (Meeting) and Event B (Training). The constraints are: - **Event A:** Duration 90 minutes. Must start between 9:00 AM and 11:00 AM. - **Event B:** Duration 60 minutes. Must finish by 3:00 PM. - **Gap:** A minimum of 2 hours is required between the end of Event A and the start of Event B. Assuming all constraints must be met, what is the earliest possible start time for Event B?

Step-by-step solution (Time Arithmetic):

1. Goal: To find the earliest start time for Event B, we must use the earliest possible schedule for Event A.
2. Calculate Earliest Finish Time for Event A:
- Earliest Start for A: 9:00 AM
- Duration of A: 90 minutes (1 hour 30 minutes)
- Earliest Finish for A: 9:00 AM + 1 hour 30 minutes = 10:30 AM.
3. Apply Minimum Gap:
- Earliest Start for B = (Earliest Finish A) + (Minimum Gap)
- Minimum Gap: 2 hours (120 minutes)
- Earliest Start for B: 10:30 AM + 2 hours = 12:30 PM.
4. Check Deadline for Event B:
- If B starts at 12:30 PM, its finish time is 12:30 PM + 60 minutes = 1:30 PM.
- The latest finish time for B is 3:00 PM. Since 1:30 PM is before 3:00 PM, the schedule is valid.
Answer: The earliest possible start time for Event B is 12:30 PM.
Key Strategy: To find the minimum time for the second event, use the minimum time for the first event, plus the mandatory gap.

time-arithmetic, gap-constraint, optimization, banking-exams

Question 13

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 14

A flow shop has 2 machines (M1 → M2). Jobs and processing times (M1, M2): - Job A: (45, 49) - Job B: (37, 37) - Job C: (23, 45) - Job D: (41, 33) - Job E: (27, 26) - Job F: (20, 36) Using Johnson's Rule, what is the minimum makespan?

Step-by-step solution (Johnson's Rule):

1. Apply Johnson's Rule:
- If M1 time < M2 time, schedule early
- If M2 time < M1 time, schedule late
2. Optimal sequence: Job D → Job E → Job F → Job C → Job B → Job A
3. Calculate makespan: 267

Answer: 267

flow-shop, multi-stage, makespan, johnsons-rule, manufacturing

Question 15

A hospital needs one doctor on-call each day for 30 days. There are 4 doctors: Dr. Smith, Dr. Brown, Dr. Lee, Dr. Patel. If the schedule is as fair as possible, how many days will each doctor be on-call?

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

on-call, availability, emergency, medical, fairness

Question 16

A student has 9 days to prepare for three exams: Mathematics, Computer Science, Chemistry. The required preparation days are: - Mathematics: 3 days - Computer Science: 1 days - Chemistry: 2 days If the student follows the optimal schedule starting today, on which day will the last exam be?

Step-by-step solution:

Timeline Planning Method:
1. Calculate total preparation time needed:
- Mathematics: 3 days
- Computer Science: 1 days
- Chemistry: 2 days
- Total: 6 days

2. Available days: 9 days
3. Extra buffer days: 3 days
4. Optimal schedule:
- Days 1-3: Prepare for Mathematics
- Day 4: Mathematics exam
- Days 5-5: Prepare for Computer Science
- Day 6: Computer Science exam
- Days 7-8: Prepare for Chemistry
- Day 9: Chemistry exam

Answer: The last exam will be on Day 9

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

academic, deadline, preparation, timeline

Question 17

In a single-elimination knockout tournament with 8 teams, how many total matches are played to determine the champion?

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

tournament, bracket, knockout, sports, competitive-exams

Question 18

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

Step-by-step solution (Deductive Logic):

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

Final Sequence: A → C → B → D → E

Answer: The task in the fourth position is D.

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

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

Question 19

Project tasks with uncertain durations (optimistic, likely, pessimistic) in days: - Design: (3, 5, 8) - Development: (2, 3, 5) - Testing: (4, 7, 9) - Deployment: (2, 5, 6) 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: (3 + 4×5 + 8)/6 = 5.2
- Development: (2 + 4×3 + 5)/6 = 3.2
- Testing: (4 + 4×7 + 9)/6 = 6.8
- Deployment: (2 + 4×5 + 6)/6 = 4.7

2. Total expected duration: 19.8 days

Answer: 19.8 days

fuzzy, uncertainty, duration, advanced, real-world

Question 20

Hospital OR scheduling with 3 operating rooms (8 hours each): - Emergency: 44 min, Priority 1 - Urgent: 54 min, Priority 2 - Elective A: 113 min, Priority 3 - Elective B: 102 min, Priority 3 - Routine: 136 min, Priority 4 Can all surgeries be completed in one day?

Step-by-step solution:

1. Total surgery time: 449 min = 7.5 hours
2. Available OR hours: 3 × 8 = 24 hours
3. Total ≤ Available → Can complete in one day

Answer: All surgeries can be scheduled within one day

surgery, operating-room, hospital, healthcare, critical

Scheduling - Intermediate Level: time allocation INTERMEDIATE

What you'll learn in this worksheet: