Scheduling Worksheet 24 - Advanced Level (schedule constraints) | 20 Questions

Question 1

Five subjects are scheduled on five different days of the week (Monday to Friday), one subject per day. The following information is given: - Chemistry is scheduled on Wednesday - History is scheduled immediately after Biology - There are exactly two classes between English and Mathematics - Biology is not on Monday On which day is History scheduled?

Step-by-step solution:

Table Method:
1. Create a timeline for Monday to Friday
2. Apply direct constraints:
- Chemistry is on Wednesday (fixed)
- Biology is not on Monday
3. Apply consecutive constraint:
- History immediately follows Biology
- Possible pairs: (Tue-Wed), (Wed-Thu), (Thu-Fri)
- Since Wednesday is occupied, options are (Tue-Wed) or (Thu-Fri)
4. Apply gap constraint:
- Two classes between English and Mathematics
5. Final schedule:
- Monday: English
- Tuesday: Mathematics
- Wednesday: Chemistry
- Thursday: Biology
- Friday: History

Answer: History is scheduled on Friday

Key Strategy: Fix direct constraints first, then work with consecutive and gap constraints.

weekly, academic, day-based, constraints

Question 2

A delivery company has vehicles with capacity 11 units. Customer demands: - C1: 7 units - C2: 8 units - C3: 7 units - C4: 8 units What is the minimum number of vehicles needed to serve all customers?

Step-by-step solution:

1. Total demand: 30
2. Vehicle capacity: 11
3. Minimum vehicles: ⌈30 ÷ 11⌉ = 3

Answer: 3 vehicles

vehicle-routing, logistics, delivery, optimization, VRP

Question 3

Computer Science courses with prerequisites: - CS102 requires CS301 - CS101 requires CS102 - CS201 requires CS101 - CS202 requires CS201 Which courses can be taken in the first semester?

Step-by-step solution:

1. Identify courses with prerequisites:
- CS102 needs CS301
- CS101 needs CS102
- CS201 needs CS101
- CS202 needs CS201

2. Courses without prerequisites (can take first): CS301

Answer: CS301

course, registration, prerequisite, academic, student

Question 4

A job shop has 3 machines. Jobs and their routes: - Job A: M1 → M3 → M2 with times 17, 33, 25 - Job B: M3 → M1 → M2 with times 39, 40, 28 - Job C: M2 → M1 → M3 with times 35, 40, 20 What is a lower bound on the minimum makespan?

Step-by-step solution:

1. Machine load bound: 97
2. Job processing bound: 107
3. Lower bound: 107

Answer: 107

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

Question 5

A factory has 5 machines. Each requires 1 day of preventive maintenance every 45 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: 1 day
3. Staggered schedule: Never have all machines down simultaneously
4. Maximum operational: 5 - 1 = 4

Answer: 4 machines

maintenance, preventive, downtime, industrial, reliability

Question 6

A passenger travels from Denver to Chicago via Atlanta. The minimum layover at Atlanta is **60 minutes**. **Flights Denver -> Atlanta:** - F1-1: Dep 8:30 AM, Arr 10:57 AM - F1-2: Dep 11:30 AM, Arr 1:57 PM - F1-3: Dep 2:30 PM, Arr 4:57 PM **Flights Atlanta -> Chicago:** - F2-1: Dep 11:00 AM, Arr 1:04 PM - F2-2: Dep 1:00 PM, Arr 3:04 PM - F2-3: Dep 3:00 PM, Arr 5:04 PM What is the minimum total elapsed time for the journey from Denver to Chicago?

1. Timeline Approach & Constraint Application (Minimum Layover: 60 min):
The fastest total time is found by checking all 9 combinations and ensuring the layover time (F2 Dep Time - F1 Arr Time) is at least the minimum required.

2. Optimal Path Calculation:
The minimum elapsed time of 334 minutes is achieved by combining F1-2 (Arr: 1:57 PM) and F2-3 (Dep: 3:00 PM, Arr: 5:04 PM).
Total Elapsed Time = Final Arrival Time - Initial Departure Time.

3. Final Answer: The minimum elapsed time is 5 hours and 34 minutes.

transportation, time-interval, optimization, CAT-level

Question 7

A student has 7 days to prepare for three exams: Chemistry, Physics, Computer Science. The required preparation days are: - Chemistry: 2 days - Physics: 2 days - Computer Science: 1 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:
- Chemistry: 2 days
- Physics: 2 days
- Computer Science: 1 days
- Total: 5 days

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

Answer: The last exam will be on Day 7

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

academic, deadline, preparation, timeline

Question 8

Round Robin scheduling with time quantum = 3: - P1: Burst time 11 - P2: Burst time 15 - P3: Burst time 15 What is the average completion time?

Step-by-step solution:

1. Round Robin simulation:
2. Completion times:
- P1: 29
- P2: 38
- P3: 41

3. Average: 108 ÷ 3 = 36.0

Answer: 36.0

preemptive, interruptible, round-robin, OS, computer-science

Question 9

Arrange the following activities in chronological order: Morning Yoga, Breakfast, Lunch Break, Office Work

Step-by-step solution:

Timeline Approach:
1. Convert all times to 24-hour format for easy comparison
- Morning Yoga: 6:00 AM
- Breakfast: 7:30 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. Breakfast at 7:30 AM
4. Office Work at 9:00 AM

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

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

daily, time-based, chronological

Question 10

Round Robin scheduling with time quantum = 4: - P1: Burst time 12 - P2: Burst time 11 - P3: Burst time 8 - P4: Burst time 6 - P5: Burst time 13 What is the average completion time?

Step-by-step solution:

1. Round Robin simulation:
2. Completion times:
- P3: 32
- P4: 34
- P1: 42
- P2: 45
- P5: 50

3. Average: 203 ÷ 5 = 40.6

Answer: 40.6

preemptive, interruptible, round-robin, OS, computer-science

Question 11

A hospital ward has 12 patients. Each nurse can handle at most 5 patients. What is the minimum number of nurses required?

Step-by-step solution:

1. Patients: 12
2. Capacity per nurse: 5
3. Minimum nurses: ⌈12 ÷ 5⌉ = 3

Answer: 3 nurses

nurse, patient, assignment, healthcare, ratio

Question 12

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.

shift-rotation, multi-person, constraints, professional

Question 13

An event runs for 8 hours. Staff needed per hour: - Hour 1: 4 - Hour 2: 5 - Hour 3: 4 - Hour 4: 4 - Hour 5: 10 (PEAK) - Hour 6: 6 - Hour 7: 4 - 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: 10 staff at hour 5
2. Staff can work multiple hours → schedule around peak
3. Minimum staff needed: 10

Answer: 10 staff

event, staff, venue, hospitality, logistics

Question 14

A football league has 10 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: 10 × (10-1) = 90
2. Maximum matches per round: 5
3. Minimum rounds: 90 ÷ 5 = 9 rounds

Answer: 9 rounds

league, fixture, home-away, sports, optimization

Question 15

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.

shift-rotation, multi-person, constraints, professional

Question 16

A traveler needs to go from City A to City D. The transport schedule is: - T1: City A to City B, Departs 06:30, Arrives 9:59 AM - T2: City A to City C, Departs 08:00, Arrives 10:53 AM - T3: City B to City D, Departs 15:00, Arrives 4:47 PM - T4: City B to City C, Departs 14:00, Arrives 3:53 PM - T5: City C to City D, Departs 10:00, Arrives 12:09 PM - T6: City C to City B, Departs 10:00, Arrives 12:31 PM - T7: City E to City D, Departs 11:00, Arrives 1:06 PM - T8: City E to City B, Departs 13:30, Arrives 3:02 PM Minimum connection time is 30 minutes. What is the earliest arrival time at City D?

Step-by-step solution:

Network Path Analysis:
1. Identify all possible routes from City A to City D:
- City A→City B -> City B→City D
- City A→City C -> City C→City D
- City A→City B -> City B→City C -> City C→City D

2. Best route found:
- T1: City A to City B (06:30 - 9:59 AM)
- Connection time: 301 minutes
- T3: City B to City D (15:00 - 4:47 PM)

Earliest arrival: 4:47 PM

Key Strategy: Enumerate all possible routes, verify connection times meet minimum requirements.

transportation, connections, optimization, railway-exams

Question 17

Arrange the following activities in chronological order: Dinner, Office Work, Morning Yoga, Breakfast

Step-by-step solution:

Timeline Approach:
1. Convert all times to 24-hour format for easy comparison
- Dinner: 8:00 PM
- Office Work: 9:00 AM
- Morning Yoga: 6:00 AM
- Breakfast: 7:30 AM

2. Arrange in chronological order:
1. Morning Yoga at 6:00 AM
2. Breakfast at 7:30 AM
3. Dinner at 8:00 PM
4. Office Work at 9:00 AM

Final Schedule: Morning Yoga -> Breakfast -> Dinner -> Office Work

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

daily, time-based, chronological

Question 18

A traveler needs to go from City A to City D. The transport schedule is: - T1: City A to City B, Departs 06:00, Arrives 9:09 AM - T2: City A to City C, Departs 07:30, Arrives 11:12 AM - T3: City B to City D, Departs 16:00, Arrives 6:38 PM - T4: City B to City C, Departs 10:30, Arrives 12:44 PM - T5: City C to City D, Departs 14:30, Arrives 4:48 PM - T6: City C to City B, Departs 14:30, Arrives 4:57 PM - T7: City E to City D, Departs 12:00, Arrives 2:31 PM - T8: City E to City B, Departs 12:30, Arrives 3:06 PM Minimum connection time is 45 minutes. What is the earliest arrival time at City D?

Step-by-step solution:

Network Path Analysis:
1. Identify all possible routes from City A to City D:
- City A→City B -> City B→City D
- City A→City C -> City C→City D
- City A→City B -> City B→City C -> City C→City D

2. Best route found:
- T2: City A to City C (07:30 - 11:12 AM)
- Connection time: 198 minutes
- T5: City C to City D (14:30 - 4:48 PM)

Earliest arrival: 4:48 PM

Key Strategy: Enumerate all possible routes, verify connection times meet minimum requirements.

transportation, connections, optimization, railway-exams

Question 19

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

Step-by-step solution:

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

Answer: No common slot available

thesis, defense, committee, academic, availability

Question 20

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

Scheduling - Advanced Level: schedule constraints ADVANCED

What you'll learn in this worksheet:

Question 1

Question 2

Question 3

Question 4

Question 5

Question 6

Question 7

Question 8

Question 9

Question 10

Question 11

Question 12

Question 13

Question 14

Question 15

Question 16

Question 17

Question 18

Question 19

Question 20