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Scheduling - Intermediate Level: priority scheduling INTERMEDIATE

Level up your scheduling skills with this comprehensive review. 20 intermediate-level problems await in Worksheet 14 of 30. Focus area: priority scheduling. Learn shift planning, time slots, schedule constraints through systematic practice. Designed for mid-level learners seeking moderate complexity with mixed patterns.

📝 Worksheet 14 of 30 • 20 questions • ⏱️ Estimated time: 20 minutes • 🎯 Intermediate level

What you'll learn in this worksheet:
Your progress through Scheduling
Worksheet 14 of 30 (46% complete)

Question 1

A factory produces Widgets with a 85% learning curve (each doubling of cumulative production reduces time by 15%). First unit takes 63 minutes. Batch sizes (in order): 40, 10, 20 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^-0.234
where exponent = log(0.85)/log(2) = -0.234

2. Calculate cumulative time using integration:
Cumulative time for N units = T_1 × N^0.765534746362977 / (learning_exponent + 1)

3. Time per batch:
Batch 1 (40 units): 34.7 minutes
Batch 2 (10 units): 25.8 minutes
Batch 3 (20 units): 24.2 minutes

4. Total time: 84.6 ≈ 85 minutes

Key Strategy: Learning curve reduces time with repetition; use cumulative average method for batch calculations.
learning-curve, improvement, batch-processing, industrial, experience

Question 2

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

Answer: 6 staff
event, staff, venue, hospitality, logistics

Question 3

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

Question 4

A factory has 3 machines. Each requires 1 day of preventive maintenance every 60 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: 1 day
3. Staggered schedule: Never have all machines down simultaneously
4. Maximum operational: 3 - 1 = 2

Answer: 2 machines
maintenance, preventive, downtime, industrial, reliability

Question 5

A passenger travels from Miami to Atlanta via Chicago. The minimum layover at Chicago is **45 minutes**. **Flights Miami -> Chicago:** - F1-1: Dep 8:00 AM, Arr 9:47 AM - F1-2: Dep 11:00 AM, Arr 12:47 PM - F1-3: Dep 2:00 PM, Arr 3:47 PM **Flights Chicago -> Atlanta:** - F2-1: Dep 11:30 AM, Arr 2:24 PM - F2-2: Dep 1:30 PM, Arr 4:24 PM - F2-3: Dep 3:30 PM, Arr 6:24 PM What is the minimum total elapsed time for the journey from Miami to Atlanta?
1. Timeline Approach & Constraint Application (Minimum Layover: 45 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 384 minutes is achieved by combining F1-1 (Arr: 9:47 AM) and F2-1 (Dep: 11:30 AM, Arr: 2:24 PM).
Total Elapsed Time = Final Arrival Time - Initial Departure Time.

3. Final Answer: The minimum elapsed time is 6 hours and 24 minutes.
transportation, time-interval, optimization, CAT-level

Question 6

Four team members (Alice, Charlie, Bob, David) must be assigned to four unique tasks (Design, Documentation, Deployment, Testing). The assignments must follow these rules: 1. Alice must handle Design. 2. Charlie cannot handle Testing. 3. Bob and David must be adjacent in (Design → Documentation → Deployment → Testing). Based on the constraints, which statement MUST be true?
No valid schedule found given the constraints. The only guaranteed assignment is: Alice must handle Design.
If the constraints cannot all be satisfied, fallback is to force rule 1's assignment.
multi-person, shift-scheduling, constraint-satisfaction, puzzle

Question 7

A traveler needs to go from City A to City E. The transport schedule is: - T1: City A to City B, Departs 07:00, Arrives 9:16 AM - T2: City A to City C, Departs 10:00, Arrives 1:37 PM - T3: City B to City E, Departs 11:30, Arrives 1:37 PM - T4: City B to City C, Departs 11:30, Arrives 2:19 PM - T5: City C to City E, Departs 11:30, Arrives 1:56 PM - T6: City C to City B, Departs 15:00, Arrives 5:41 PM - T7: City D to City E, Departs 14:30, Arrives 4:52 PM - T8: City D to City B, Departs 10:00, Arrives 12:09 PM Minimum connection time is 45 minutes. What is the earliest arrival time at City E?
Step-by-step solution:

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

2. Best route found:
- T1: City A to City B (07:00 - 9:16 AM)
- Connection time: 134 minutes
- T3: City B to City E (11:30 - 1:37 PM)

Earliest arrival: 1:37 PM

Key Strategy: Enumerate all possible routes, verify connection times meet minimum requirements.
transportation, connections, optimization, railway-exams

Question 8

A clinic operates for 4 hours with 15-minute appointment slots. If 12 patients need appointments, how many can be accommodated?
Step-by-step solution:

1. Total slots available: (4 × 60) ÷ 15 = 16
2. Patients: 12
3. All patients can be scheduled

Answer: All 12 patients can be scheduled
clinic, appointment, doctor, healthcare, wait-time

Question 9

**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 10

A hospital needs to schedule 5 staff for 7 days (Thursday, Sunday, 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: 2 points
- Bob: 8 points
- Carol: 0 points
- Emma: 2 points
- Frank: 3 points

4. Fairness gap: 8 - 0 = 8

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 11

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

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

Answer: No common slot available
thesis, defense, committee, academic, availability

Question 12

A factory produces Widgets with a 90% learning curve (each doubling of cumulative production reduces time by 9%). First unit takes 89 minutes. Batch sizes (in order): 10, 20, 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): 74.0 minutes
Batch 2 (20 units): 56.9 minutes
Batch 3 (30 units): 50.1 minutes

4. Total time: 180.9 ≈ 181 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

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 14

Four team members (Charlie, David, Frank, Alice) must be assigned to four unique tasks (Testing, Design, Deployment, Documentation). The assignments must follow these rules: 1. Charlie must handle Testing. 2. David cannot handle Documentation. 3. Frank and Alice must be adjacent in (Testing → Design → Deployment → Documentation). Based on the constraints, which statement MUST be true?
No valid schedule found given the constraints. The only guaranteed assignment is: Charlie 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 15

A machine can process up to 6 jobs simultaneously as a batch. Each batch takes 44 minutes. If 14 jobs need to be processed, what is the minimum total time required?
Step-by-step solution:

1. Jobs per batch: 6
2. Number of batches: ⌈14 ÷ 6⌉ = 3
3. Total time: 3 × 44 = 132 minutes

Answer: 132 minutes
batch, parallel-machines, capacity, manufacturing, optimization

Question 16

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

Answer: 6 staff
event, staff, venue, hospitality, logistics

Question 17

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: - V attends in March - W attends on the 15th - Exactly two people attend between S and R - U attends in the same month as P In which month does S 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:
- V in March (Mar 5 or Mar 15)
- W on 15th (any month, date 15)
- Two people between S and R
(If S at position 1, R at position 4)
- U and P in same month

3. Systematic placement:
- Place V at Mar 5 (satisfies March constraint)
- Place W at Mar 15 (satisfies 15th constraint)
- For 'two between' constraint: If S at Jan 5, R at Mar 15
- U and P together: May 5 & May 15

4. Verification:
All constraints satisfied with S 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 18

A machine needs to process 4 jobs. Processing times: - Job A: 75 minutes - Job C: 61 minutes - Job E: 77 minutes - Job B: 69 minutes The machine breaks down at 114 minutes and takes 39 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 B → Job A → Job E
2. Simulate processing with breakdown:
- Job C: 0 → 61
- Job B: Starts at 61, breakdown at 114 (53 min completed), repair 39 min, resume 16 min → completes at 169
- Job A: 169 → 244
- Job E: 244 → 321

3. Total makespan: 321 minutes
4. Delay caused by breakdown: 39 minutes

Answer: 321 minutes

Key Strategy: Simulate the timeline, account for breakdown during active job processing.
manufacturing, breakdown, recovery, simulation, reliability

Question 19

A conference needs to schedule 5 sessions across 3 time slots and 3 rooms. Each room can hold one session per slot. The constraints are: - Prof. Wilson can only speak at 10:00-11:00 - Cybersecurity and Machine Learning cannot be in the same time slot - Prof. Wilson and Dr. Chen must speak in consecutive time slots - Robotics must be in Hall B Which speaker presents the Cybersecurity session?
Step-by-step solution:

Scheduling Grid Analysis:
1. Fix direct constraints:
- Prof. Wilson at 10:00-11:00
- Robotics in Hall B
2. Apply consecutive constraint: Prof. Wilson and Dr. Chen in consecutive slots
3. Apply conflict constraint: Cybersecurity and Machine Learning not together

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

Answer: Prof. Wilson presents Cybersecurity

Key Strategy: Use a grid to solve the assignment problem and satisfy all constraints sequentially.
multi-constraint, professional, CSP, CAT-level

Question 20

A machine can process up to 5 jobs simultaneously as a batch. Each batch takes 31 minutes. If 20 jobs need to be processed, what is the minimum total time required?
Step-by-step solution:

1. Jobs per batch: 5
2. Number of batches: ⌈20 ÷ 5⌉ = 4
3. Total time: 4 × 31 = 124 minutes

Answer: 124 minutes
batch, parallel-machines, capacity, manufacturing, optimization
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