Scheduling - Beginner-Intermediate Level: time slots BEGINNER-INTERMEDIATE

Comprehensive race against clock worksheet covering 20 beginner-intermediate-level scheduling problems. Worksheet 8 of 30 emphasizes time slots. Master schedule logic, time allocation, day scheduling through detailed explanations. Difficulty: building on fundamentals with moderate challenges. Tailored for developing preparation.

📝 Worksheet 8 of 30 • 20 questions • ⏱️ Estimated time: 20 minutes • 🎯 Beginner-intermediate level

What you'll learn in this worksheet:

Analyze complex schedule logic patterns systematically
Develop analytical thinking for time allocation problems
Learn step-by-step race against clock approaches
Understand the logic behind day scheduling solutions
Apply critical thinking to time slots challenges

Your progress through Scheduling

Worksheet 8 of 30 (26% complete)

Question 1

A machine needs to process 4 jobs. Processing times: - Job B: 84 minutes - Job A: 56 minutes - Job C: 34 minutes - Job E: 60 minutes The machine breaks down at 66 minutes and takes 23 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 C → Job A → Job E → Job B
2. Simulate processing with breakdown:
- Job C: 0 → 34
- Job A: Starts at 34, breakdown at 66 (32 min completed), repair 23 min, resume 24 min → completes at 113
- Job E: 113 → 173
- Job B: 173 → 257

3. Total makespan: 257 minutes
4. Delay caused by breakdown: 23 minutes

Answer: 257 minutes

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

manufacturing, breakdown, recovery, simulation, reliability

Question 2

A conference needs to schedule 6 sessions across 3 time slots and 3 rooms. Each room can hold one session per slot. The constraints are: - Dr. Smith can only speak at 9:00-10:00 - Robotics and Machine Learning cannot be in the same time slot - Prof. Wilson and Dr. Smith must speak in consecutive time slots - Cloud Computing must be in Hall A Which speaker presents the IoT session?

Step-by-step solution:

Scheduling Grid Analysis:
1. Fix direct constraints:
- Dr. Smith at 9:00-10:00
- Cloud Computing in Hall A
2. Apply consecutive constraint: Prof. Wilson and Dr. Smith in consecutive slots
3. Apply conflict constraint: Robotics and Machine Learning not together

4. Final Schedule:
9:00-10:00:
- Hall A: Cloud Computing by Dr. Smith
- Hall B: Robotics by Prof. Wilson
- Hall C: Cybersecurity by Prof. Jones
10:00-11:00:
- Hall A: Machine Learning by Prof. Garcia
- Hall B: Data Science by Dr. Taylor
- Hall C: IoT by Prof. Brown
11:00-12:00:
- Hall A: (empty)
- Hall B: (empty)
- Hall C: (empty)

Answer: Prof. Brown presents IoT

Key Strategy: Use a grid to solve the assignment problem and satisfy all constraints sequentially.

multi-constraint, professional, CSP, CAT-level

Question 3

Five patients need appointments. Their preferences are: - Patient A: Dr. Patel at 10:00 AM - Patient B: Dr. Kumar at 10:00 AM - Patient C: Dr. Patel at 10:30 AM - Patient D: Dr. Shah at 10:00 AM - Patient E: Dr. Patel at 11:00 AM Each doctor can see one patient per 30-minute slot. If all preferences are honored, how many patients need to be rescheduled?

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.

appointment, conflict-resolution, multi-person

Question 4

A factory produces Widgets with a 90% learning curve (each doubling of cumulative production reduces time by 9%). First unit takes 90 minutes. Batch sizes (in order): 20, 40, 10 units. What is the TOTAL production time for all batches (in minutes, rounded to nearest minute)?

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): 67.3 minutes
Batch 2 (40 units): 51.8 minutes
Batch 3 (10 units): 47.7 minutes

4. Total time: 166.8 ≈ 167 minutes

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

learning-curve, improvement, batch-processing, industrial, experience

Question 5

A delivery company has vehicles with capacity 20 units. Customer demands: - C1: 7 units - C2: 5 units - C3: 6 units - C4: 3 units What is the minimum number of vehicles needed to serve all customers?

Step-by-step solution:

1. Total demand: 21
2. Vehicle capacity: 20
3. Minimum vehicles: ⌈21 ÷ 20⌉ = 2

Answer: 2 vehicles

vehicle-routing, logistics, delivery, optimization, VRP

Question 6

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

Step-by-step solution:

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

Answer: No common slot available

thesis, defense, committee, academic, availability

Question 7

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 8

**Data Sufficiency Question** A project has 4 phases: Planning, Design, Development, Testing. **Question:** On which day does Development start? **Statement (1):** Testing starts exactly 5 days after Design ends. **Statement (2):** Planning takes 3 days and ends before Design starts. **Options:** A. Statement (1) ALONE is sufficient, but statement (2) alone is NOT sufficient B. Statement (2) ALONE is sufficient, but statement (1) alone is NOT sufficient C. BOTH statements TOGETHER are sufficient, but NEITHER alone is sufficient D. EACH statement ALONE is sufficient E. Statements (1) and (2) TOGETHER are NOT sufficient

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.

data-sufficiency, GMAT, CAT, reasoning, test-prep

Question 9

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 10

A job shop has 3 machines. Jobs and their routes: - Job A: M3 → M2 → M1 with times 32, 38, 24 - Job B: M1 → M3 → M2 with times 33, 17, 39 - Job C: M2 → M3 → M1 with times 14, 16, 26 What is a lower bound on the minimum makespan?

Step-by-step solution:

1. Machine load bound: 91
2. Job processing bound: 94
3. Lower bound: 94

Answer: 94

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

Question 11

A delivery company has vehicles with capacity 12 units. Customer demands: - C1: 7 units - C2: 4 units - C3: 7 units - C4: 8 units - C5: 3 units What is the minimum number of vehicles needed to serve all customers?

Step-by-step solution:

1. Total demand: 29
2. Vehicle capacity: 12
3. Minimum vehicles: ⌈29 ÷ 12⌉ = 3

Answer: 3 vehicles

vehicle-routing, logistics, delivery, optimization, VRP

Question 12

A factory produces Widgets with a 90% learning curve (each doubling of cumulative production reduces time by 9%). First unit takes 81 minutes. Batch sizes (in order): 10, 40, 30 units. What is the TOTAL production time for all batches (in minutes, rounded to nearest minute)?

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 (10 units): 67.3 minutes
Batch 2 (40 units): 49.1 minutes
Batch 3 (30 units): 43.0 minutes

4. Total time: 159.4 ≈ 159 minutes

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

learning-curve, improvement, batch-processing, industrial, experience

Question 13

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 14

A hospital needs one doctor on-call each day for 30 days. There are 5 doctors: Dr. Patel, Dr. Jones, Dr. Smith, Dr. Brown, Dr. Lee. 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 15

An event runs for 8 hours. Staff needed per hour: - Hour 1: 3 - Hour 2: 3 - Hour 3: 5 - Hour 4: 3 - Hour 5: 4 (PEAK) - Hour 6: 5 - Hour 7: 3 - Hour 8: 6 What is the minimum number of staff needed if staff can work multiple consecutive hours?

Step-by-step solution:

1. Identify peak demand: 6 staff at hour 8
2. Staff can work multiple hours → schedule around peak
3. Minimum staff needed: 6

Answer: 6 staff

event, staff, venue, hospitality, logistics

Question 16

A hospital needs to schedule 5 staff for 7 days (Thursday, Friday, Monday...). Each day has 3 shifts: Morning, Evening, Night. Undesirable shifts (higher weight = more undesirable): - Weekend Night: weight 3 - Weekend Evening: weight 2 - Any Night: weight 1 After creating a fair schedule, what is the fairness gap (difference between max and min undesirable weights assigned to any staff)?

Step-by-step solution (Fairness Scheduling):

1. Total shifts to assign:
- 7 days × 3 shifts = 21 shifts
2. Shifts per person: 21 ÷ 5 = 4 with 1 extra shifts
3. Undesirable weight distribution:
- Alice: 3 points
- Bob: 3 points
- Carol: 1 points
- David: 4 points
- Frank: 4 points

4. Fairness gap: 4 - 1 = 3

Key Strategy: Fair scheduling aims to minimize the maximum difference in undesirable shift assignments across all staff.

fairness, shift-scheduling, workforce, equity, human-resources

Question 17

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 1 - Package: 15 min on Line 1 **Product Y:** - Cut: 20 min on Line 3 - Assemble: 60 min on Line 3 - Package: 20 min on Line 3 **Product Z:** - Cut: 40 min on Line 2 - Assemble: 30 min on Line 1 - 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: 90 minutes
- Line 2: 40 minutes
- Line 3: 155 minutes

2. Identify bottleneck: The line with maximum load = Line 3
3. Bottleneck load: 155 minutes

Answer: Line 3 (155 minutes)

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

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

Question 18

In a round-robin tournament with 4 teams, each round consists of disjoint matches (no team plays twice in a round). What is the minimum number of rounds needed?

Step-by-step solution:

1. **Model as edge coloring of complete graph K_4
2. Vizing's theorem: χ'(K_n) = n-1 for even n, n for odd n
3. For 4 teams: 3 colors/rounds needed

Answer: 3 rounds

edge-coloring, match-scheduling, graph-theory, advanced, olympiad

Question 19

A company has 7 employees working in 2 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: 7 employees × 10 days = 70 days
2. Verification: Each employee cycles through all shifts

Answer: 70 days

rotation, cyclic, shift-pattern, workforce, healthcare

Question 20

An event runs for 8 hours. Staff needed per hour: - Hour 1: 3 - Hour 2: 6 - Hour 3: 5 - Hour 4: 6 - Hour 5: 7 (PEAK) - Hour 6: 6 - Hour 7: 6 - Hour 8: 5 What is the minimum number of staff needed if staff can work multiple consecutive hours?

Step-by-step solution:

1. Identify peak demand: 7 staff at hour 5
2. Staff can work multiple hours → schedule around peak
3. Minimum staff needed: 7

Answer: 7 staff

event, staff, venue, hospitality, logistics

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