1. Industrial Systems Engineering Dept. İzmir University of Economics
Scheduling
Mahmut Ali GÖKÇE
Industrial Systems Engineering Dept.
İzmir University of Economics

2. Topics Scheduling Scheduling in high-volume systems
Scheduling in intermediate-volume systems
Scheduling in low-volume systems
Gantt charts
Sequencing
Priority rules and performance measures
Single machine scheduling
Proof that WSPT is optimal for minimizing weighted completion time
Sequencing jobs through two work centers
Cyclic scheduling in service systems

3. Scheduling
Establishing the timing of the use of resources such as equipment, facilities, and human activities.

4. Manufacturers
Schedule production, developing schedules for workers, equipment, purchases, maintenance, etc.
Effective scheduling can yield cost savings and increases in productivity

5. Hospitals
Schedule admissions, surgery, nursing assignments, support services such as meal preparation, security, maintenance, cleaning.
Effective scheduling can save lives and improve patient care.

6. Educational Institutions
Schedule classrooms, instruction and students.
Effective scheduling can reduce the need for expansion of facilities and satisfy constraints of people.

7. Scheduling Decisions Dependent on earlier decisions such as
Capacity of the system
Equipment selections
Selection and training of workers
Design of products and services
May aim at achieving a tradeoff among conflicting goals: Efficient utilization vs. minimization of waiting times, inventory, process times.

8. Scheduling Operations
High-volume systems (flow systems)
Intermediate-volume systems
Low volume systems (job shop)

9. Scheduling in High-Volume Systems
Standardized equipment and activities that provide identical or highly similar operations on customers or products as they pass through the system.
Perform best with a high, uniform output.

10. Scheduling in High-Volume Systems
Goal: Smooth rate of flow of goods or customers
Production Ex: Production of autos, personal computers, radios and television, toys, appliances,
Process Ex: Petroleum refining, sugar refining, waste treatment
Service Ex: Cafeteria lines

11. Scheduling in High-Volume Systems
Line balancing: Allocating the required tasks to workstations so that they satisfy sequencing constraints and are balanced wrt equal work times among stations. (Ch 6)

12. Scheduling in High-Volume Systems
Goals: Maximum utilization of equipment and personnel, highest possible rate of output
Handling disruptions can reduce output: Equipment failures, material shortages, accidents, absences.

13. Scheduling in High-Volume Systems
For success:
Product and product design
Preventive maintenance
Rapid repair when breakdowns occur
Optimal product mixes
Minimization of quality problems
Reliability and timing of supplies

14. Scheduling in Intermediate-Volume Systems
Between standardized type of output of flow-shop and made-to-order output of job shops.
Periodically shift from one job to another
Run sizes are larger than job shop.
Ex: Production of canned foods, baked goods, paint, cosmetics
Issues: Run size and timing of jobs and the sequence in which jobs should be processed.

15. Demand
Setup cost
production rate
production Quantity
Setup cost may depend on the order in which jobs are processed: similar jobs may require less setup change between them.
Makes sequencing more complex, since setup costs have to be observed/estimated for every sequence combination
usage rate
Holding cost

16. Scheduling in Low-Volume Systems (Job-shop Scheduling)
Products are made to order
Orders usually differ considerably in terms of processing requirements, materials needed, processing time, and processing sequence and setup.
Issues:
How to distribute the load among work centers (loading)
What job processing to use

17. Scheduling in Low-Volume Systems (Job-shop Scheduling)
Gantt charts: Organize and visually display the actual or intended use of resources in a time framework.

18. Gantt Charts
Ex: Load chart: Helps a manager rework loading assignments to better utilize work centers

19. Gantt Charts
Ex: Schedule chart: Shows which jobs are on schedule and which are behind or late.

20. Infinite Loading vs. Finite Loading
Infinite loading: Assigns jobs to work centers without regard to the capacities of the work centers.
Queues may form
Finite loading: Considers capacities of the work centers.
More complex

21. General Approaches to Scheduling
Forward scheduling: Scheduling ahead, from some point in time.
“How long will it take to complete this job?”
Backward scheduling: Scheduling by working backwards from the due dates.
“When is the latest time the job can be started and still be completed by the due date?”

22. ::: Skip the assignment method :::

23. Sequencing
Loading determines the machines or work centers for jobs, but not the order in which they are processed.
Sequencing is concerned with determining job processing order.

24. Priority Rules
Priority rules are simple heuristics used to select the order (sequence) in which the jobs will be processed.
Typical assumption: Job setup cost and time are independent of the processing sequence.
Job processing times and due dates are important pieces of information.
Due dates may be the result of
Delivery times promised to customers
MRP processing
Managerial decisions

25. Priority Rules FCFS (first come, first served)
SPT (shortest processing time)
EDD (earliest due date)
CR (critical ratio)
S/O (slack per operation)
Rush

26. Performance Measures Job flow time Job lateness Tardiness Makespan:
Total time needed to complete a group of jobs
Average number of jobs

27. EXAMPLE 2
Determine the sequence and performance measures for the given group of jobs, arriving in the given sequence:
Job Sequence
Processing Time
Due Date A 2 7 B 8 16 C 4 D 10 17 E 5 15 F 12 18

28. EXAMPLE 2 FCFS solution: A-B-C-D-E-F Makespan = 41 days
Seq.
Proc. Time
Due Date
Flow Time
Days Tardy A 2 7 B 8 16 10 C 4 14 D 17 24 E 5 15 29 F 12 18 41 23
120 54
Makespan = 41 days
Average Flow Time = 120/6=20 days
Average Tardiness = 54/6=9 days
Average Number of Jobs = 120/41=2.93 jobs/day

29. EXAMPLE 2 SPT solution: A-C-E-B-D-F Makespan = 41 days
Seq.
Proc. Time
Due Date
Flow Time
Days Tardy A 2 7 C 4 6 E 5 15 11 B 8 16 19 3 D 10 17 29 12 F 18 41 23
108 40
Makespan = 41 days
Average Flow Time = 108/6=18 days
Average Tardiness = 40/6=6.67 days
Average Number of Jobs = 108/41=2.63 jobs/day

30. EXAMPLE 2 EDD solution: C-A-E-B-D-F Makespan = 41 days
Seq.
Proc. Time
Due Date
Flow Time
Days Tardy C 4 A 2 7 6 E 5 15 11 B 8 16 19 3 D 10 17 29 12 F 18 41 23
110 38
Makespan = 41 days
Average Flow Time = 110/6=18.33 days
Average Tardiness = 38/6=6.33 days
Average Number of Jobs = 110/41=2.68 jobs/day

31. EXAMPLE 2 CR solution: C-F-A-E-B-D Makespan = 41 days
Seq.
Proc. Time
Due Date
Flow Time
Days Tardy C 4 F 12 18 16 A 2 7 11 E 5 15 23 8 B 31 D 10 17 41 24
133 58
Makespan = 41 days
Average Flow Time = 133/6=22.17 days
Average Tardiness = 58/6=9.67 days
Average Number of Jobs = 133/41=3.24 jobs/day

32. EXAMPLE 2 Comparison of the 4 rules: Rule Average Flow Time
Average Tardiness
Average Number of Jobs
FCFS
20.00
9.00
2.93
SPT
18.00
6.67
2.63
EDD
18.33
6.33
3.68 CR
22.17
9.67
3.24

33. SPT SPT is always superior (optimal) in terms of
Minimizing flow time
Minimizing the average number of jobs (and work-in-process inventory)
Completion time
Disadvantage: Makes long jobs wait

34. FCFS FCFS is widely used in service systems since it is Fair Simple
And since we typically can not estimate processing times of individual jobs.

35. EDD & CR EDD typically performs good in minimizing lateness
Disadvantage: Does not consider job processing times.
CR typically performs good in minimizing job tardiness

36. Theorem: For 1||wjCj the WSPT rule is optimal.
Proof (by contradiction): Suppose a schedule S, which is not WSPT, is optimal. In S, there must be at least two adjacent jobs, say job i followed by job k, such that
wi / pi < wk / pk .
Assume job i starts its processing at time t.
Consider a new schedule S` where jobs i and k are exchanged and all other jobs remain in their original position. The total weighted completion times of the jobs processed before and after i and k are unchanged in S`.

37. Theorem: For 1||wjCj the WSPT rule is optimal.
Schedule S i k t
t + pi + pk
Schedule S` k i t
t + pi + pk

38. Proof (by contradiction) continued…
Total weighted completion times of jobs i and k :
Under S: (t+pi)wi + (t+pi+pk)wk
>
Under S`: (t+pk)wk + (t+pk+pi)wi
Contradicts the optimality of S and proves the theorem.

39. Sequencing Jobs through Two Work Centers

40. Sequencing Jobs through Two Work Centers
Johnson’s rule
Minimizes makespan for a group of jobs to be processed on two machines or at two work centers
Assumptions:
Job time must be known and constant for each job at each work center
Job times must be independent of the job sequence
All jobs must follow the same two-step work sequence
Job priorities can not be used
All units of a job must be completed at the first work center before a job moves on to the second work center

41. Johnson’s Rule List the jobs and their times at each work center
Select the job with the shortest time. If the shortest time is at the first work center, schedule that job first; if the time is at the second work center, schedule the job last. Break ties arbitrarily.
Eliminate the job and its time from further consideration
Repeat steps 2 and 3, working toward the center of the sequence, until all jobs have been sequenced.

42. EXAMPLE 4 D 2nd 3rd 4th 5th 6th Processing times (hours) Job
Work Center 1
Work Center 2 A 5 B 4 3 C 8 9 D 2 7 E 6 F 12 15 D
2nd
3rd
4th
5th
6th

43. EXAMPLE 4 D 2nd 3rd 4th 5th B Processing times (hours) Job
Work Center 1
Work Center 2 A 5 B 4 3 C 8 9 E 6 F 12 15 D
2nd
3rd
4th
5th B

44. EXAMPLE 4 D 2nd 3rd 4th A B Processing times (hours) Job Work Center 15 C 8 9 E 6 F 12 15 D
2nd
3rd
4th A B

45. EXAMPLE 4 D E 3rd 4th A B Processing times (hours) Job Work Center 18 9 E 6 F 12 15 D E
3rd
4th A B

46. EXAMPLE 4 D E C 4th A B Processing times (hours) Job Work Center 18 9 F 12 15 D E C
4th A B

47. D E C F A B

48. Cyclical Scheduling in Service Systems
Employees must be assigned to work shifts or time slots, and have days off, on a repeating (cyclic) basis.
Day
Mon
Tue
Wed
Thu
Fri
Sat
Sun
Staff needed 2 4 3 6 5

49. Cyclical Scheduling in Service Systems
ALGORITHM:
1. Make the first worker’s assignment such that the two days with the lowest need are designated as days off. Circle those days. In case of a tie, pick the pair with the lowest adjacent requirement.
Day
Mon
Tue
Wed
Thu
Fri
Sat
Sun
Staff needed 2 4 3 6 5
Worker1

50. 2. Subtract 1 from each day’s requirement, except for the circled days
2. Subtract 1 from each day’s requirement, except for the circled days. Assign the next employee, again using the two lowest consecutive days as days off:
Day
Mon
Tue
Wed
Thu
Fri
Sat
Sun
Staff needed 2 4 3 6 5
Worker1

51. 3. Repeat the preceding step for each additional worker until all staffing requirements have been met. Don’t subtract from a value of zero.

52. Gant chart is a graphical representation of tasks over a specific period of time.  In its correct form, the Gant chart should represent the tasks, resources, time frame and required founding needed by the project manager to complete its intended purpose.