Boost your speed and accuracy with this adaptive style 📈 worksheet. Worksheet 15 of 30 presents 20 intermediate-level scheduling problems. Focus on conflict resolution while practicing schedule logic, time allocation, day scheduling. Difficulty: moderate complexity with mixed patterns. Perfect for mid-level test takers.
A factory has 5 machines. Each requires 2 day of preventive maintenance every 90 days. If maintenance is staggered, what is the maximum number of machines that can be operational at any time?
Step-by-step solution:
1. Total machines: 5 2. Maintenance duration: 2 day 3. Staggered schedule: Never have all machines down simultaneously 4. Maximum operational: 5 - 1 = 4
Answer: 4 machines
Question 2
A clinic operates for 4 hours with 15-minute appointment slots. If 8 patients need appointments, how many can be accommodated?
Step-by-step solution:
1. Total slots available: (4 × 60) ÷ 15 = 16 2. Patients: 8 3. All patients can be scheduled
Answer: All 8 patients can be scheduled
Question 3
A hospital needs to schedule 5 staff for 7 days (Sunday, Friday, Saturday...). 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: 1 points - Bob: 6 points - Carol: 3 points - Emma: 3 points - Frank: 2 points
4. Fairness gap: 6 - 1 = 5
Key Strategy: Fair scheduling aims to minimize the maximum difference in undesirable shift assignments across all staff.
Question 4
An event runs for 6 hours. Staff needed per hour:
- Hour 1: 3
- Hour 2: 4
- Hour 3: 3
- Hour 4: 8 (PEAK)
- Hour 5: 4
- Hour 6: 4
What is the minimum number of staff needed if staff can work multiple consecutive hours?
Step-by-step solution:
1. Identify peak demand: 8 staff at hour 4 2. Staff can work multiple hours → schedule around peak 3. Minimum staff needed: 8
Answer: 8 staff
Question 5
A company has 7 employees working in 2 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: 7 employees × 7 days = 49 days 2. Verification: Each employee cycles through all shifts
Answer: 49 days
Question 6
Four employees need to be scheduled for three shifts over three days. The constraints are:
- Each employee works exactly one shift per day
- No employee works the same shift two days in a row
- Alice works Morning shift on Monday
- Bob cannot work Night shift
- Charlie works Evening shift on Tuesday
Who works the Evening shift on Wednesday?
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.
Question 7
A school needs to schedule 6 courses. The following courses have overlapping students and cannot be scheduled at the same time:
- Physics conflicts with Chemistry
- Physics conflicts with English
- Physics conflicts with CS
- Physics conflicts with History
- Chemistry conflicts with History
- Chemistry conflicts with CS
- Chemistry conflicts with English
- Math conflicts with English
- Math conflicts with History
What is the minimum number of time slots needed to schedule all courses without conflicts?
Key Strategy: The chromatic number of the conflict graph gives the minimum slots needed.
Question 8
Four employees need to be scheduled for three shifts over three days. The constraints are:
- Each employee works exactly one shift per day
- No employee works the same shift two days in a row
- Alice works Morning shift on Monday
- Bob cannot work Night shift
- Charlie works Evening shift on Tuesday
Who works the Evening shift on Wednesday?
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.
Question 9
**Data Sufficiency Question**
Six lectures are scheduled from Monday to Saturday, one per day.
**Question:** Which subject is on Thursday?
**Statement (1):** Physics is on Wednesday, two days after Chemistry.
**Statement (2):** Mathematics is on Friday, immediately after Biology.
**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: Physics is on Wednesday, two days after Chemistry. This gives partial information but not enough to determine the answer uniquely.
Step 2 - Analyze Statement (2) alone: Mathematics is on Friday, immediately after Biology. This also gives partial information insufficient by itself.
Step 3 - Combine statements: Together, they provide enough constraints to solve uniquely.
Conclusion: Either statement alone is sufficient.
Key Strategy: Test each statement independently first, then combine only if neither alone works.
Question 10
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:
- Prof. Garcia can only speak at 10:00-11:00
- Cybersecurity and Blockchain cannot be in the same time slot
- Dr. Smith and Prof. Garcia must speak in consecutive time slots
- Robotics must be in Hall A
Which speaker presents the AI Ethics session?
Step-by-step solution:
Scheduling Grid Analysis: 1. Fix direct constraints: - Prof. Garcia at 10:00-11:00 - Robotics in Hall A 2. Apply consecutive constraint: Dr. Smith and Prof. Garcia in consecutive slots 3. Apply conflict constraint: Cybersecurity and Blockchain not together
4. Final Schedule: 9:00-10:00: - Hall A: Robotics by Dr. Lee - Hall B: Cybersecurity by Dr. Taylor - Hall C: Data Science by Dr. Smith 10:00-11:00: - Hall A: Machine Learning by Prof. Garcia - Hall B: Blockchain by Prof. Jones - Hall C: AI Ethics by Prof. Wilson 11:00-12:00: - Hall A: (empty) - Hall B: (empty) - Hall C: (empty)
Answer: Prof. Wilson presents AI Ethics
Key Strategy: Use a grid to solve the assignment problem and satisfy all constraints sequentially.
Question 11
Round Robin scheduling with time quantum = 4:
- P1: Burst time 13
- P2: Burst time 5
- P3: Burst time 11
- P4: Burst time 12
- P5: Burst time 10
What is the average completion time?
Arrange the following activities in chronological order: Lunch Break, Breakfast, Evening Walk, Office Work
Step-by-step solution:
Timeline Approach: 1. Convert all times to 24-hour format for easy comparison - Lunch Break: 1:00 PM - Breakfast: 7:30 AM - Evening Walk: 6:00 PM - Office Work: 9:00 AM
2. Arrange in chronological order: 1. Lunch Break at 1:00 PM 2. Evening Walk at 6:00 PM 3. Breakfast at 7:30 AM 4. Office Work at 9:00 AM
Final Schedule: Lunch Break -> Evening Walk -> Breakfast -> Office Work
Key Strategy: Convert all times to 24-hour format and arrange from earliest to latest.
Question 13
A clinic operates for 4 hours with 30-minute appointment slots. If 11 patients need appointments, how many can be accommodated?
Answer: 3 patients will need to wait or be rescheduled
Question 14
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 1
**Product Y:**
- Cut: 20 min on Line 3
- 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 2
- 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: 60 minutes - Line 2: 70 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.
Question 15
In a double round-robin tournament (each pair plays twice), 6 teams compete. How many total matches will be played?
A factory has 3 machines. Each requires 2 day of preventive maintenance every 30 days. If maintenance is staggered, what is the maximum number of machines that can be operational at any time?
Step-by-step solution:
1. Total machines: 3 2. Maintenance duration: 2 day 3. Staggered schedule: Never have all machines down simultaneously 4. Maximum operational: 3 - 1 = 2
Answer: 2 machines
Question 17
A factory produces Widgets with a 85% learning curve (each doubling of cumulative production reduces time by 15%).
First unit takes 61 minutes.
Batch sizes (in order): 10, 20, 40 units.
What is the TOTAL production time for all batches (in minutes, rounded to nearest minute)?
Key Strategy: Learning curve reduces time with repetition; use cumulative average method for batch calculations.
Question 18
Computer Science courses with prerequisites:
- CS101 requires CS401
- CS201 requires CS401
- CS301 requires CS401
- CS102 requires CS201
Which courses can be taken in the first semester?
2. Courses without prerequisites (can take first): CS401
Answer: CS401
Question 19
A bus route takes 36 minutes one-way. Peak frequency: 1 bus every 15 minutes. What is the minimum number of buses needed to maintain this frequency in both directions?
A machine needs to process 4 jobs. Processing times:
- Job B: 76 minutes
- Job E: 54 minutes
- Job A: 80 minutes
- Job C: 85 minutes
The machine breaks down at 87 minutes and takes 22 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 E → Job B → Job A → Job C 2. Simulate processing with breakdown: - Job E: 0 → 54 - Job B: Starts at 54, breakdown at 87 (33 min completed), repair 22 min, resume 43 min → completes at 152 - Job A: 152 → 232 - Job C: 232 → 317
3. Total makespan: 317 minutes 4. Delay caused by breakdown: 22 minutes
Answer: 317 minutes
Key Strategy: Simulate the timeline, account for breakdown during active job processing.
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